library(dplyr)
library(kableExtra)
library(knitr)
library(nortest)
library(gridExtra)
library(grid)
library(pracma)
knitr::opts_chunk$set(echo = TRUE)
options(knitr.table.format = "html")

不使用科學記號

options(scipen=999)

Question 01

(a) The mid-Square Method

Write a computer program using the Mid-Square Method using 6 digits to generate 10,000 random numbers ranging over [0, 999999]. Use the Kolmogorov-Smirnov Goodness-of-fit test to see if the random numbers that you create are uniformly distributed. (Note: You must notify the initial seed number used, and you may adapt 0.05 as the α value. Also, you may find warning messages for conducting the Goodness-of-fit test, and comment on the Goodness-of-fit test. )

# N 為模擬次數, x 為起始值(亂數決定)#
set.seed(2)
x <- floor(runif(1)*10^6)
N <- 10000
u <-  matrix(0,N/5,5)
for(i in 1:N/5){
  for( j in 1:5){
    y <-  x^2 %>% as.vector()
    while(nchar(y) != 12){
      y <- paste0(0,y)}
    x <- substr(y,4,9) %>% as.numeric()
    u[i,j]  <- substr(y,4,9)}}
kable(u) %>% kable_styling() %>% scroll_box(height = "200px")
325851 178874 995907 830752 148885
783455 801737 782217 863435 519999
252456 734031 801508 415074 286425
011957 142969 440134 717937 433535
743076 161941 224887 574162 662002
008510 072420 244656 856558 691607
208110 309772 958691 088433 820395
148411 025824 666878 726266 462302
650199 758739 684870 046916 201111
012395 153636 604020 840160 868825
983541 352898 536998 366852 580389
849423 519432 809602 455398 387338
057774 337835 132487 552805 593368
990989 059198 504403 422386 409932
836879 366460 292931 808570 785444
829358 834692 710734 142818 396981
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
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906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
202965 194791 943533 254522 781448
857655 572099 297265 366480 307590
617139 860545 537697 118063 938871
325596 012755 162690 468036 057697
383260 888227 947203 193523 451151
777163 982328 968299 602953 552320
356033 759497 835693 382790 528184
958218 181735 027610 762312 119585
187422 127006 130524 036514 333272
722015 305660 428035 213961 779309
094878 001834 003363 011309 127893
285481 499401 401358 088244 787003
702682 761993 633332 109422 973174
135779 435936 040196 615718 108655
529539 411552 375048 661002 923644
881095 328399 845903 551885 577053
358356 419022 579436 746078 632382
660976 889272 804689 524386 980676
611608 064345 140279 678197 951170
478754 205392 185873 548772 150707
328943 203497 411029 944838 718846
537224 609626 643859 554411 371556
057382 292693 669192 817932 012756
978337 143285 530591 526809 527722
300572 343527 010799 116618 599757
070225 931550 785402 856301 251402
322517 017215 296356 826878 727226
356619 177111 368306 649309 602177
373722 668133 401705 366907 620746
067634 574357 885963 930437 713010
805909 489316 430147 026441 699126
118238 980224 839090 072028 188032
990164 424746 409164 415178 372771
906993 636302 880235 813655 034459
725416 228373 154227 785967 744125
724368 708999 679582 831694 714909
712599 797334 741507 832631 274382
739571 965264 734589 620998 638516
053861 901007 813614 967740 520707
162715 476171 738821 856470 540860
490509 599079 895648 185339 350544
708459 914154 677535 053676 881112
358356 419022 579436 746078 632382 
hist(as.numeric(u)/1000000, main = " The mid-square method ",xlab= "Data",breaks = 400)

P <- ks.test(as.numeric(u)/1000000,'punif')
## Warning in ks.test(as.numeric(u)/1000000, "punif"): ties should not be
## present for the Kolmogorov-Smirnov test
P$p.value
## [1] 0
print(ifelse(P$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Reject H0; This is not an uniform distribution"
# P value < alpha = 0.05, reject H0: This is uniform distribution #

(b) The combination of 3 multiplicative congruential generators

Consider the combination of 3 multiplicative congruential generators, i.e.,
\(u_i=\frac{x_i}{30269}+\frac{y_i}{30307}+\frac{z_i}{30323}\)(mod 1)

with \(x_i\) =171\(x_{i-1}\) (mod30269), \(y_i\) =172 \(y_{i-1}\) (mod30307), \(z_i\) =170\(z_{i-1}\) (mod30323). Use both the \(x^2\) and Kolmogorov-Smirnov Goodness-of-fit tests to check if the data are from U(0,1) distribution.

# N 為模擬次數, (x, y, z) 為任意給定值 #
N = 10000
x <- 5; y <- 8; z <- 34
u <- NULL
for(i in 1:N){
  x <- (x * 171 ) %% 30269
  y <- (y * 172 ) %% 30307
  z <- (z * 170 ) %% 30323
  temp <- (x/30269 + y/30307 + z/30323) %% 1
  u <- c(u, temp)}
u <- matrix(u,ncol=5)
kable(u) %>% kable_styling() %>% scroll_box(height = "200px")
0.2642632 0.0711310 0.7140377 0.2520377 0.6910384
0.0437878 0.0006039 0.8295787 0.7819919 0.3591055
0.8924169 0.7349981 0.7490125 0.6607122 0.7366934
0.0218728 0.9673492 0.9635072 0.8553598 0.0160317
0.2485370 0.1582377 0.3026961 0.6884113 0.9252578
0.8729845 0.3005099 0.7603075 0.7612790 0.8984499
0.6573878 0.7279885 0.2506518 0.1723766 0.6488055
0.9872906 0.1663427 0.5972115 0.3035168 0.2507577
0.6322797 0.2335053 0.4782565 0.7062420 0.6420172
0.4218715 0.4258172 0.8251468 0.2326660 0.6183557
0.0696737 0.9186240 0.0462698 0.4906426 0.7631394
0.2928525 0.5184204 0.6882126 0.9790809 0.6583009
0.6765218 0.3065429 0.3645330 0.5827639 0.8929791
0.7115257 0.2991902 0.4702463 0.5554481 0.2054306
0.2023078 0.4250717 0.2106141 0.2455437 0.4925026
0.3386329 0.4029453 0.7658363 0.7335160 0.9352451
0.6576473 0.7683270 0.4766342 0.2161075 0.4588521
0.0154808 0.5958452 0.9738957 0.4830718 0.4966042
0.6834891 0.0907364 0.3444026 0.6376581 0.4185623
0.6599609 0.4637434 0.7023002 0.4290144 0.6695855
0.0998550 0.4010286 0.9839336 0.5664158 0.0406067
0.0721740 0.8027664 0.2228565 0.7907606 0.9210549
0.6174791 0.5493942 0.8858871 0.7601255 0.7659902
0.2928809 0.5122330 0.3733210 0.3046149 0.2666587
0.7898141 0.6926156 0.2159328 0.3767159 0.7814739
0.0406814 0.3863045 0.2204841 0.9165451 0.4606702
0.7618875 0.2080782 0.1026168 0.6303403 0.1911192
0.4353153 0.1708234 0.1434545 0.6908620 0.1303007
0.8888387 0.5602311 0.3723289 0.0699098 0.7763139
0.9722680 0.5554529 0.7689364 0.6466920 0.7449645
0.0165935 0.1250444 0.1817676 0.8580492 0.8599557
0.4669939 0.7480385 0.7728390 0.9760258 0.8683144
0.7775744 0.3394928 0.3666485 0.9467443 0.7058011
0.6464760 0.7580202 0.3233667 0.1662241 0.7378524
0.5872307 0.0804440 0.4633915 0.7990292 0.8310060
0.0347555 0.7679282 0.7041215 0.6122917 0.0150460
0.7603908 0.6334614 0.0900602 0.8598091 0.6262000
0.4764247 0.7872627 0.2675649 0.7469033 0.4172224
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0.0463785 0.2948513 0.4233525 0.0416346 0.2019290
0.7600580 0.4539947 0.6780236 0.1161482 0.1971724
0.6948629 0.9634845 0.7370385 0.0413442 0.2872697
0.0184617 0.2010322 0.1810285 0.9765839 0.0175242
0.9748881 0.1941192 0.4033415 0.6075149 0.1516195
0.9375700 0.8416996 0.6425394 0.3211000 0.7068152
0.9444609 0.5429957 0.2181008 0.8568852 0.9601441
0.6211771 0.7719459 0.7418136 0.7096113 0.0649156
0.3765111 0.1690699 0.0587290 0.1031397 0.5948072
0.4023512 0.2506021 0.6856612 0.5690039 0.3134688
0.3588546 0.6323919 0.6235390 0.3334897 0.8966653
0.6357665 0.2432632 0.8676035 0.4867448 0.6681890
0.7076785 0.5754086 0.5991517 0.5082227 0.2947922
0.7943223 0.5192528 0.5194354 0.4941086 0.6930069
0.2894028 0.1249889 0.1078086 0.3986197 0.4287144
0.9228871 0.3715353 0.9617810 0.3166686 0.7908775
0.6030095 0.9537682 0.9958348 0.6171619 0.4223289
0.4664308 0.9592533 0.9869175 0.4820325 0.4831841
0.5564606 0.9802133 0.2035735 0.3102689 0.4231449
0.7537597 0.1630226 0.0472432 0.2682713 0.3842194
0.5233953 0.1466860 0.5272697 0.2220132 0.6808557
0.4317697 0.4095395 0.9874287 0.6789081 0.1315577
0.1011349 0.1553323 0.3000135 0.0301768 0.5863315
0.8158566 0.9902222 0.2908716 0.9916222 0.1814111
0.7581830 0.5545537 0.5384836 0.0892575 0.6606723
0.7868458 0.0644111 0.1314392 0.6707796 0.9800863
0.1728483 0.2175640 0.4367371 0.8102136 0.2793303
0.4121994 0.9838617 0.4958283 0.3913825 0.9419512
0.1924176 0.4919633 0.3970736 0.2536245 0.4304753
0.9401282 0.7274800 0.8424539 0.6469732 0.8873747
0.5629508 0.0725039 0.2789188 0.6001516 0.6736856
0.5159773 0.4078377 0.3601519 0.9901775 0.1878607
0.1137230 0.1419848 0.4738255 0.1722401 0.3580220
0.0697818 0.0721236 0.2299485 0.8676342 0.0902411
0.8323698 0.5536295 0.0633515 0.9923250 0.7875911
0.1810271 0.9339114 0.4522868 0.8756288 0.3844846
0.3865158 0.3097594 0.2493550 0.0171484 0.2839435
0.2952634 0.1592293 0.6571583 0.4275339 0.7060936
0.2890029 0.2677973 0.2139833 0.8420630 0.5083632
0.4779159 0.0760181 0.7436501 0.7239669 0.6386504
0.0972679 0.1258604 0.0291783 0.6031549 0.0894728
0.1554308 0.4387621 0.4784671 0.2558719 0.2950370
0.9449053 0.8744346 0.3392744 0.7862080 0.2973152
0.6445620 0.4727309 0.7296716 0.4816120 0.6642019
0.9039175 0.3389841 0.9587158 0.1715007 0.3246062
0.6161272 0.8512951 0.3592115 0.6321854 0.0507153
0.3670785 0.5695557 0.9547610 0.2755416 0.7486131
0.8558900 0.8539778 0.3100242 0.0175213 0.1794662
0.7016238 0.0044356 0.1606027 0.3554882 0.7385204
0.8063731 0.8244028 0.5092033 0.2907042 0.0008299
0.1240647 0.0332868 0.2910119 0.3240325 0.2181300
0.9864364 0.9040059 0.0361975 0.2842740 0.6710754
0.6266324 0.9437259 0.1748146 0.8146495 0.8297192
0.5991665 0.0333946 0.2520695 0.3058311 0.1927628
0.2111053 0.2486570 0.3769592 0.4354740 0.0124480
0.0330294 0.5150719 0.1609652 0.5360640 0.0643975
0.7477765 0.0749829 0.7421342 0.0559096 0.7441084
0.0133913 0.1720000 0.7991940 0.5081535 0.9306289
0.9283228 0.6445998 0.6970208 0.6356201 0.5844337
0.8184866 0.7972048 0.4648356 0.3056473 0.1916226
0.7299974 0.1670689 0.2381330 0.7186886 0.1919457
0.5179211 0.9017721 0.7041861 0.6840190 0.8040958
0.6276346 0.8774607 0.1879882 0.9198135 0.2928164
0.0890511 0.8237631 0.5810288 0.1778060 0.2767724
0.4705275 0.3639818 0.0592634 0.6791765 0.7689874
0.8559066 0.0333443 0.8717344 0.2963203 0.9971285
0.6521810 0.1015764 0.7124537 0.3960171 0.7222274
0.8276268 0.8244433 0.7222435 0.8558789 0.9371977
0.0273563 0.3745588 0.5151505 0.8849845 0.9653683
0.0595485 0.1852580 0.7015081 0.8354735 0.3115112
0.8955350 0.3981255 0.0909711 0.9315021 0.5320463
0.4763787 0.7740738 0.1203649 0.3093250 0.5604425
0.2087459 0.9215359 0.1849930 0.4869022 0.7811225
0.8949867 0.8637084 0.4918154 0.9784730 0.5609511
0.9949014 0.2200435 0.8069310 0.2536747 0.0014324
0.3698287 0.9416510 0.0979469 0.1431324 0.7026315
0.1391032 0.7737633 0.3735596 0.6124447 0.5305674
0.2788114 0.8995517 0.9364969 0.7423358 0.9828322
0.8466829 0.3076771 0.4426110 0.9051737 0.2699022
0.9069480 0.0293071 0.6483581 0.2526094 0.7507819
0.7062820 0.3740131 0.0315178 0.7967327 0.2108915
0.5796922 0.6143681 0.4693605 0.9217809 0.0377228
0.3408757 0.4712506 0.3801204 0.9912314 0.0115194
0.3650851 0.7559628 0.9060654 0.4146970 0.9003860
0.1968729 0.8984992 0.2395786 0.0934573 0.0867776
0.0144887 0.1516010 0.0627039 0.4154813 0.5394467
0.6657052 0.9707449 0.1204122 0.4981994 0.2642718
0.8332563 0.3040652 0.0315750 0.7595130 0.7610582
0.3419948 0.1981970 0.1691565 0.9159351 0.8701060
0.3887535 0.9373380 0.6420612 0.9082610 0.2640703
0.8001726 0.5948276 0.7485018 0.7401804 0.4460510
0.1693894 0.8001088 0.2382195 0.6017756 0.8723611
0.3504890 0.4767856 0.9553782 0.7509672 0.3110872
0.7917635 0.0300165 0.8993516 0.9777189 0.1068504
0.3274581 0.9525412 0.7753758 0.0747968 0.0410720
0.0261998 0.4040625 0.0078792 0.0927604 0.5008888
0.2012436 0.1854190 0.5707201 0.6423308 0.4182352
0.6455568 0.1825887 0.8615652 0.4750520 0.1721947
0.7690682 0.8545246 0.6873186 0.0812381 0.8343027
0.0261047 0.8721772 0.4512186 0.8414524 0.2714838
0.1508710 0.7480685 0.0175726 0.1747833 0.0754294
0.4516891 0.7663001 0.8832736 0.7444727 0.3527271
0.5234005 0.7051982 0.4382496 0.1273925 0.1422303
0.2291997 0.8647419 0.8682420 0.0307138 0.0246089
0.4356116 0.6199665 0.7139253 0.1140595 0.7298192
0.2627743 0.1360300 0.7914424 0.8456574 0.0143827
0.5326963 0.0007395 0.9558081 0.5294534 0.8579996
0.8095950 0.6305768 0.3155755 0.7510804 0.8035717
0.0594631 0.0683521 0.9602401 0.6153009 0.7052255
0.1726139 0.3101112 0.3940828 0.6563805 0.1656554
0.5375380 0.3750748 0.1147255 0.2726777 0.7880659
0.5937348 0.0313343 0.8689982 0.9299393 0.3610039
0.1188469 0.2180888 0.4314961 0.1206954 0.9282924
0.8318868 0.4145573 0.9985504 0.3023612 0.4330053
0.1236710 0.2244524 0.3396329 0.2186499 0.5217667
0.7363152 0.8551591 0.9899931 0.1734172 0.7226142
0.4118923 0.2697023 0.4588313 0.4843741 0.1464625
0.0019103 0.1290129 0.1465379 0.0786841 0.3201288
0.5110554 0.9854821 0.6499469 0.9665435 0.1651902
0.4144948 0.4651146 0.6982143 0.6117811 0.5313664
0.3784096 0.0434096 0.7174369 0.5153199 0.4679122
0.3169152 0.5289410 0.9524300 0.0887638 0.2532637
0.5437301 0.8599169 0.8767987 0.3997077 0.7804379
0.7590813 0.0405789 0.0546117 0.6141715 0.2191507
0.8099851 0.2894114 0.6221151 0.3959905 0.6667355
0.4458065 0.7924230 0.1931385 0.4978207 0.2516071
0.9531968 0.8143003 0.6689831 0.2270434 0.0900375
0.9616451 0.1977844 0.2964236 0.0756666 0.0851844
0.3974922 0.1772078 0.5512156 0.4361852 0.9575318
0.7953207 0.0211303 0.9105513 0.4263936 0.9828047
0.2107296 0.5337992 0.4540871 0.7008013 0.0487345
0.8831548 0.2710966 0.5970616 0.8637687 0.8959278
0.5993139 0.3366956 0.6012032 0.4872798 0.4551529
0.7765560 0.7090268 0.4249890 0.1200184 0.5789308
0.5155743 0.1422878 0.9795000 0.5595786 0.1509995
0.0792509 0.0688126 0.6350603 0.4483721 0.0516453
0.4668457 0.6784940 0.0372176 0.6447032 0.7031273
0.3392785 0.4258072 0.3631033 0.9881923 0.0641549
0.1079120 0.4109652 0.3011489 0.6842931 0.7435470
0.5861508 0.3270649 0.0734065 0.4294337 0.5205312
0.7177778 0.0602058 0.4514055 0.5914789 0.5185751
0.4463946 0.4973861 0.8843189 0.6217523 0.9162321
0.5450783 0.3298035 0.9960966 0.0159491 0.2219478
0.6047435 0.8096090 0.4223060 0.1392953 0.4481289
0.7348944 0.6331269 0.8041932 0.5235158 0.4759070
0.0847227 0.1752158 0.8839453 0.1975059 0.3128624
0.9466861 0.2077144 0.7619910 0.5765363 0.4861613
0.8734571 0.0169025 0.5213064 0.0173138 0.4424608
0.6672945 0.6499754 0.6711322 0.2558654 0.3243044
0.2554454 0.8586047 0.5682756 0.8922655 0.6636506
0.0218404 0.7037057 0.2065064 0.2921336 0.7495687
0.2857361 0.4222085 0.5944025 0.9287440 0.9389040
0.1598085 0.2122840 0.4287806 0.1817614 0.4537382
0.3460875 0.4820717 0.2712238 0.7337711 0.7411728
0.7690563 0.7954132 0.6937981 0.9011367 0.0063677
0.9412083 0.0790322 0.8979334 0.7035713 0.0778096
0.1332566 0.9843809 0.2532528 0.1859826 0.0048569
0.9196117 0.0804397 0.7304962 0.4037158 0.4176690
0.5263212 0.5938508 0.9366062 0.9992829 0.2063393
0.3423929 0.1101609 0.9174431 0.3004182 0.8324580
0.8632543 0.3028107 0.1143679 0.4096634 0.1497958
0.7057358 0.3723647 0.1782384 0.2935551 0.4947131
0.7656632 0.0109114 0.8174364 0.9451001 0.5324968
0.3003782 0.8638048 0.8173005 0.1905125 0.8182253
0.8646783 0.2235210 0.0617919 0.0997807 0.2009482
0.5753826 0.5292517 0.0709130 0.4708593 0.1369352
0.8474091 0.1300952 0.8655580 0.5202799 0.0301907
0.2337738 0.4273672 0.4641895 0.7043252 0.1601409
0.5433995 0.8850733 0.3869131 0.5809051 0.2255277 
h = hist(u,main='The combination of 3 multiplicative congruential generators',xlab = 'Data')

# Kolmogorov-Smirnov #
p1 <- ks.test(u,'punif')
p1$p.value
## [1] 0.4986677
print(ifelse(p1$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"
# Chisq test #
p2 <- chisq.test(h$counts,p = rep(1/length(h$counts),length(h$counts)))
p2$p.value
## [1] 0.6036862
print(ifelse(p2$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"

(c) \(U_{n+1} = (\)π$ +\(U_n)^5\)(mod 1)

The calculators use\(U_{n+1}\) = (\(π\) +\(U_n)^5\)(mod 1) to generate random numbers between 0 and 1. Compare the result with those in (a) & (b), and discuss your finding based on the comparison.

# N 為模擬次數 #
set.seed(5)
u0 <- runif(1,0,1)
N <- 10000
U <- NULL
for(i in 1:N){
  if(i == 1){U[i] <- ((pi+u0)^5) %% 1}
  else{
  U[i] <- ((pi+U[i-1])^5) %% 1}}
U <- matrix(U,ncol=5)
kable(U) %>% kable_styling() %>% scroll_box(height = "200px")
0.7800198 0.2104336 0.2509217 0.6426022 0.3396884
0.5201614 0.1916207 0.3745711 0.0101948 0.3231052
0.3333751 0.4485716 0.4591238 0.0173464 0.2600915
0.7034972 0.4466243 0.2637168 0.5619817 0.4806738
0.4897873 0.8308077 0.9129379 0.7954903 0.5940757
0.4783253 0.1566188 0.7283229 0.9601421 0.5143944
0.5751069 0.2945645 0.9754687 0.0150751 0.1654807
0.2302561 0.0326390 0.8691938 0.4327302 0.5661952
0.8534044 0.2501458 0.8813652 0.4033350 0.7681776
0.6121895 0.8609058 0.7253748 0.8051547 0.6002387
0.3243522 0.2019822 0.6743288 0.6275746 0.5353445
0.1591864 0.8835088 0.0905644 0.7239044 0.0955715
0.8160724 0.5356622 0.7461706 0.0316660 0.4869078
0.9461941 0.3860244 0.1759823 0.7565267 0.9786567
0.4086913 0.2697929 0.8867826 0.0686754 0.4559852
0.0472377 0.0128004 0.8392871 0.9623728 0.6303922
0.7289502 0.3050818 0.7589202 0.1755126 0.5714620
0.6792116 0.4086857 0.8354056 0.6023505 0.7594936
0.2803707 0.0427347 0.8944548 0.6076820 0.4992157
0.2203026 0.4074401 0.9799060 0.8602483 0.7187434
0.4582162 0.0539580 0.2571994 0.3586244 0.2856422
0.5012374 0.2180597 0.5477166 0.3816411 0.8455590
0.4969047 0.0275588 0.4792354 0.8843361 0.6593958
0.6910132 0.6796531 0.3568641 0.6218276 0.3827826
0.9338949 0.7508431 0.0618447 0.9420429 0.7643312
0.3404973 0.5261470 0.3503848 0.6248773 0.1151362
0.9174292 0.7316382 0.2268555 0.0058113 0.3599526
0.8105877 0.6997274 0.6599699 0.8605491 0.3792282
0.2370146 0.3777979 0.9820789 0.7443549 0.0278362
0.2390393 0.9298502 0.3955778 0.1040100 0.8195544
0.5597028 0.7723635 0.7069316 0.1444960 0.2249423
0.6543878 0.5013150 0.2500906 0.1742825 0.4298205
0.1698664 0.5652436 0.8243719 0.8582918 0.0325923
0.1960293 0.8694483 0.1697250 0.8521611 0.2264340
0.1767442 0.2106804 0.1110161 0.0297874 0.3887098
0.3484442 0.3474121 0.0484012 0.8052354 0.3521842
0.7857510 0.0205581 0.3309053 0.7254976 0.5660133
0.3175863 0.1642978 0.9053569 0.8117227 0.5963117
0.2963719 0.8592457 0.5228169 0.6223360 0.6942856
0.2938023 0.0735800 0.7239155 0.4521082 0.4702750
0.5015613 0.5749306 0.0440405 0.3901751 0.6942336
0.7821282 0.0620731 0.0792862 0.4911185 0.4139247
0.0161776 0.4706444 0.6346436 0.6366189 0.2167255
0.9804947 0.0086805 0.8843349 0.8947953 0.1783036
0.1068126 0.2708999 0.6202926 0.4316778 0.2947370
0.7020730 0.7629080 0.4036537 0.5449704 0.1528516
0.9344153 0.4600737 0.0567901 0.9391876 0.0706806
0.0585003 0.0624627 0.6972824 0.6602148 0.0285876
0.5930467 0.6758708 0.7193073 0.2378492 0.1988511
0.5129723 0.7266405 0.9119802 0.7830709 0.9305560
0.8961268 0.0904053 0.4348513 0.1339649 0.7424229
0.2000955 0.6593438 0.1364370 0.0736595 0.9033768
0.7058255 0.3284617 0.4987144 0.6174177 0.8697655
0.0375768 0.1313737 0.2785184 0.5291846 0.6212097
0.7644013 0.5845221 0.9517322 0.4846250 0.3224523
0.1966924 0.2591169 0.1614235 0.0026020 0.7898094
0.5883279 0.8285694 0.1456584 0.2890885 0.1550857
0.9347028 0.3729720 0.8524678 0.2277202 0.3883108
0.4553922 0.2381340 0.4199353 0.2168936 0.0423957
0.1338273 0.9688035 0.0358479 0.2851986 0.2331881
0.9944476 0.3252086 0.8822591 0.5396521 0.7516974
0.3857530 0.7774312 0.8965348 0.0416340 0.5071194
0.0596672 0.4629797 0.7424433 0.8419440 0.6927300
0.2053495 0.5114518 0.9265986 0.0996359 0.7877139
0.9919862 0.5411485 0.3120483 0.7241679 0.6548037
0.7883570 0.4167735 0.3442934 0.3258825 0.6017502
0.4215497 0.4967794 0.7139427 0.2643500 0.0181166
0.3357355 0.5812344 0.9683213 0.3388682 0.9456558
0.4267321 0.0959588 0.6370823 0.7210399 0.6572763
0.5246313 0.6996032 0.3670155 0.8386682 0.1731637
0.3613303 0.2425497 0.7093996 0.9820864 0.1824529
0.4155772 0.8570507 0.9606385 0.4064266 0.8212756
0.5384235 0.2646267 0.7178018 0.2505064 0.3456223
0.9142713 0.5250727 0.2406675 0.0995908 0.6958211
0.5312612 0.7601310 0.6240820 0.6992783 0.1337532
0.3719908 0.2375317 0.2058584 0.8888925 0.9518098
0.4899740 0.5760611 0.3114229 0.6202979 0.2704047
0.6406644 0.1411821 0.8996226 0.4089634 0.4272737
0.0254407 0.2460845 0.8539880 0.2634052 0.9638284
0.6127603 0.1799852 0.3556903 0.7034562 0.2416593
0.8912322 0.3171800 0.1832678 0.4450228 0.2734487
0.7110662 0.0055312 0.3189792 0.5046212 0.4936959
0.7949135 0.7231211 0.2943989 0.4818466 0.8840865
0.2674339 0.1576314 0.9172278 0.6042626 0.2941239
0.4176033 0.8940665 0.5373236 0.4880342 0.7255889
0.1622262 0.4648189 0.9062869 0.9555675 0.9137763
0.6235960 0.0654633 0.7706759 0.5551714 0.8616264
0.7173817 0.2600371 0.5227918 0.4125684 0.1269945
0.7747104 0.4442156 0.7012904 0.1337758 0.0786145
0.2583566 0.8370180 0.0806536 0.9648256 0.2733600
0.3203011 0.9128546 0.3710450 0.6586766 0.4333854
0.2430214 0.6158200 0.7696367 0.6323940 0.9382549
0.1664458 0.9356254 0.3061766 0.5997347 0.3675614
0.1437104 0.7296371 0.1816802 0.0414459 0.1231861
0.7165155 0.4502963 0.3498305 0.7454197 0.9101772
0.8146964 0.8806269 0.8149310 0.3186236 0.0030488
0.2595022 0.7588100 0.5469312 0.0394492 0.5074663
0.0862868 0.7078044 0.7519729 0.7218503 0.0001970
0.4181147 0.2078502 0.8236454 0.7410941 0.1156658
0.5726674 0.5633468 0.2714417 0.3923976 0.6579679
0.9058325 0.0805837 0.1304361 0.2222261 0.8935877
0.1608395 0.3333882 0.0468680 0.6881652 0.8299150
0.7982580 0.7130324 0.5378571 0.8659542 0.0457164
0.2897297 0.9630924 0.3950420 0.6965214 0.9431919
0.6720209 0.1965794 0.2876901 0.8933701 0.2234338
0.6467710 0.5181824 0.2599566 0.5414817 0.4618594
0.2940349 0.5562894 0.3903127 0.7232975 0.5664210
0.6635851 0.4571282 0.5981540 0.3543490 0.9815880
0.7645572 0.5883123 0.4942478 0.1807792 0.6859243
0.3781378 0.9196068 0.3661552 0.8006544 0.4584253
0.1905959 0.7692348 0.0578610 0.1803468 0.6768017
0.8164347 0.8360490 0.2579698 0.5373042 0.7155847
0.3907199 0.6995049 0.0619476 0.8885279 0.7840752
0.9150625 0.1355654 0.4045540 0.1393140 0.3258720
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0.9568473 0.4386628 0.9801310 0.3522247 0.9257474
0.3595702 0.2609859 0.5818600 0.5962117 0.1467233
0.0918675 0.0797507 0.6969477 0.5966014 0.4746514
0.4578110 0.8846439 0.3559627 0.9770366 0.4273477
0.1611301 0.0262653 0.3870900 0.1231677 0.0238279
0.9710717 0.0277717 0.0953457 0.8997305 0.8023536
0.5660428 0.7870436 0.3629592 0.9978785 0.2341258
0.6242370 0.8561198 0.6429693 0.4141409 0.3601548
0.3617254 0.0752522 0.3866724 0.3894707 0.5311723
0.7130756 0.4693698 0.7716514 0.9433762 0.2910646
0.0106826 0.9243512 0.6661018 0.4799393 0.5979753
0.2580925 0.2374544 0.4062530 0.9620537 0.3195147
0.1438779 0.5256234 0.1129090 0.7230653 0.6784667
0.8140936 0.2579755 0.1089303 0.0954051 0.4869141
0.5213783 0.0657641 0.8826533 0.3955292 0.9841453
0.4280110 0.4191267 0.4133577 0.6688392 0.3861280
0.5621367 0.3856532 0.7638334 0.2874620 0.3500093
0.9413326 0.9824178 0.5358608 0.1022889 0.9477684
0.6375066 0.8855151 0.5675940 0.1905553 0.6082051
0.7996218 0.1714747 0.0910411 0.7914674 0.3771932
0.9339482 0.1639621 0.0063512 0.1371450 0.4661218
0.4140390 0.6587602 0.1255308 0.9076195 0.1682731
0.3080321 0.7196388 0.2440018 0.5606993 0.2390129
0.4940844 0.2803477 0.8101350 0.5899956 0.5424645
0.2233935 0.2045038 0.6848265 0.5500732 0.6280170
0.4360261 0.4616795 0.2810591 0.6649342 0.1704263
0.0980929 0.4147236 0.6924320 0.1797491 0.5328216
0.8735018 0.8553457 0.4656466 0.1734776 0.7929385
0.4673312 0.0870303 0.7658696 0.3719890 0.8984473
0.1933087 0.8218452 0.9068369 0.4886620 0.2875277
0.4914785 0.0482566 0.5092417 0.5004986 0.1476692
0.9502045 0.2560717 0.5756592 0.8466674 0.0279361
0.0186135 0.7957951 0.7574441 0.0608272 0.8699822
0.1933844 0.3263959 0.1283865 0.8149167 0.9017512
0.5382471 0.6355240 0.8736805 0.5293903 0.6955294
0.7525391 0.7798955 0.6995237 0.6714319 0.8175372
0.4744094 0.3731962 0.1559821 0.0240315 0.7443791
0.2204056 0.4091469 0.9179669 0.9045342 0.1315847
0.5239796 0.4091886 0.5405405 0.4198675 0.7056018
0.7728082 0.4423381 0.8578045 0.9813260 0.7925012
0.0232486 0.2865964 0.2286207 0.3073345 0.3746356
0.5116587 0.5041555 0.7974121 0.0002872 0.5084458
0.7253711 0.0703834 0.2711034 0.1596101 0.8689926
0.6701874 0.8704073 0.9009166 0.0676634 0.6210871
0.7095584 0.4523366 0.5806785 0.4252990 0.1996200
0.1353130 0.5806519 0.5621442 0.3638666 0.4094899
0.8502285 0.5366745 0.9483598 0.3276823 0.6818738
0.5738502 0.3119870 0.4353756 0.5665723 0.1211017
0.0320477 0.3007052 0.5653903 0.1245908 0.7276730
0.9501049 0.3282838 0.0078802 0.7088362 0.2469049
0.8790734 0.0024161 0.8769972 0.3413137 0.7204711
0.7272452 0.1982709 0.0171925 0.5178242 0.2057486
0.7675442 0.5694390 0.4853444 0.2350595 0.2424754
0.8604086 0.8389829 0.6247779 0.9667157 0.8082916
0.5641963 0.3769859 0.9057999 0.3483629 0.4392267
0.8813295 0.3072184 0.1171279 0.7254286 0.7244400
0.6785593 0.9181682 0.4816051 0.7345511 0.6297906
0.5854740 0.8139113 0.3961531 0.9824992 0.9627790
0.1770283 0.2981887 0.1573133 0.0032366 0.7518197
0.5207570 0.5641728 0.7056529 0.5993098 0.6476451
0.8688844 0.8591154 0.8485164 0.6253006 0.1946722
0.4810088 0.9066039 0.4020297 0.4318647 0.3354100
0.8824625 0.1963256 0.7752348 0.6973099 0.1888544
0.1631295 0.3606223 0.8753256 0.7492659 0.7440546
0.1620016 0.8827984 0.8392714 0.7173694 0.7616138
0.4898244 0.6036238 0.7392117 0.7611176 0.9571115
0.5105690 0.8593806 0.2555335 0.3813088 0.7323096
0.7554864 0.2463400 0.4372679 0.6283200 0.4555240
0.8683656 0.3482614 0.1159551 0.4764274 0.2442029
0.8102422 0.6501067 0.8208041 0.9474073 0.9422840
0.8156047 0.7354261 0.7643890 0.1033660 0.9601467
0.3726233 0.9705389 0.1823169 0.7873084 0.0216074
0.9721662 0.8041956 0.7383049 0.1716496 0.6892163
0.1325397 0.4645232 0.2275629 0.2693401 0.9972292
0.2540543 0.8153806 0.1155364 0.7063100 0.4613298
0.4531157 0.0978709 0.5850952 0.5685080 0.1200253
0.2308480 0.7512392 0.8116570 0.9565100 0.1181764
0.2360793 0.9808870 0.5420546 0.8837818 0.0731670
0.6299958 0.6733284 0.2505786 0.8940326 0.3543564
0.1703847 0.0305507 0.1473643 0.4198283 0.1862776
0.5077963 0.1914635 0.8495143 0.9497406 0.1613741
0.2927719 0.3515695 0.6673748 0.3686212 0.1162852
0.7843721 0.1082100 0.7450935 0.9271679 0.0067113
0.6784797 0.4807272 0.9463457 0.0918904 0.3023675
0.5006856 0.6400337 0.6203592 0.4703087 0.4964691
0.0112037 0.3802415 0.4703318 0.7229261 0.3094128
0.5154558 0.8068819 0.7425894 0.9401357 0.4724551
0.1141881 0.7248563 0.0928310 0.9753983 0.5516102
0.8269899 0.0947731 0.9847170 0.7680133 0.0934764
0.4124089 0.0487269 0.2146177 0.4082769 0.3380360
0.0065672 0.4995716 0.8394345 0.7181134 0.1108021
0.2316106 0.0315457 0.9440068 0.5863971 0.9286684
0.7295257 0.6955575 0.3581809 0.0681010 0.1495200
0.3251135 0.8480179 0.0488470 0.6574943 0.1123875
0.7087172 0.7704437 0.5618191 0.4002070 0.8165273 
h <- hist(U,main = 'Simulation',xlab = 'Data')

p1 <- ks.test(U,'punif')
p1$p.value
## [1] 0.6239733
print(ifelse(p1$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"
p2 <- chisq.test(h$counts,p = rep(1/length(h$counts),length(h$counts)))
p2$p.value
## [1] 0.6088307
print(ifelse(round(p2$p.value, digits = 3)<= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"

Result and Discuss

Question 02

There are several ways for checking the goodness-of-fit for empirical data. In specific, there are a lot of normality tests available in R. Generate a random sample of size 10, 50, and 100 from N(0,1) and t-distribution (with degrees 10 and 20) in R. You may treat testing random numbers from t-distribution as the power. For a level of significance α = 0.05 test, choose at least four normality tests in R (“nortest” module) to check if this sample is from N(0,1). Tests used can include the Kolmogorov-Smirnov test and the Cramer-von Mises test. Note that you need to compare the differences among the tests you choose.

使用的常態檢定 6種

-Kolmogorov-Smirnov test -Anderson-Darling test -Shapiro-Francia test

-Cramer-von Mises test -Lilliefors test -Pearson Chi-Square test

pvalue <- function(t1){
  kp <- ks.test(t1,'pnorm')
  ap <- ad.test(t1)
  sp <- sf.test(t1)
  cp <- cvm.test(t1)
  lp <- lillie.test(t1)
  pp <- pearson.test(t1)
  AA <- matrix(c(kp$p.value,ap$p.value,sp$p.value,cp$p.value,lp$p.value,pp$p.valuw),6,1)
  return(AA)
}

t distribution with df=10

# 抽取樣本數，實驗模擬1000次，計算各檢定的型二誤差機率 #
set.seed(5)
sample <- c(10,50,100)
PPP <- NULL
for(j in 1:3){
  PPP1 <- NULL
  for(i in 1:1000){
    t1 <- rt(sample[j],10)
    PPP1 <- cbind(PPP1,pvalue(t1))}
  PPP <- rbind(PPP,PPP1)}

PoWeR_10 <- apply(PPP<0.05,1,sum) /1000
PoWeR_10 <- matrix(PoWeR_10,3,6,byrow = T)
colnames(PoWeR_10) <- c("KS.test","AD.test","SF.test","CVM.test","Lillie.test","Pearson.test")
rownames(PoWeR_10) <- c("n = 10","n = 50","n = 100")
# 繪圖表格 #
# 1000次模擬實驗，計算Type 2 error的次數，成功拒絕H0的次數(P value < 0.05 ) #
kable(PoWeR_10, "html") %>%
  kable_styling(bootstrap_options = "striped", full_width = F) %>% 
  add_header_above(c(" Number of sample" = 1, "t distribution with df=10"=6))
Number of sample
t distribution with df=10
KS.test AD.test SF.test CVM.test Lillie.test Pearson.test
n = 10 0.052 0.061 0.070 0.058 0.057 0.052
n = 50 0.063 0.135 0.195 0.120 0.112 0.063
n = 100 0.049 0.156 0.279 0.130 0.099 0.049

t distribution with df=20

# 抽取樣本數，實驗模擬1000次，計算各檢定的型二誤差機率 #
sample <- c(10,50,100)
PPP <- NULL
for(j in 1:3){
  PPP1 <- NULL
  for(i in 1:1000){
    t1 <- rt(sample[j],20)
    PPP1 <- cbind(PPP1,pvalue(t1))}
  PPP <- rbind(PPP,PPP1)}

PoWeR_20 <- apply(PPP<0.05,1,sum)/1000
PoWeR_20 <- matrix(PoWeR_20,3,6,byrow = T)
colnames(PoWeR_20) <- c("KS.test","AD.test","SF.test","CVM.test","Lillie.test","Pearson.test")
rownames(PoWeR_20) <- c("n = 10","n = 50","n = 100")
# 繪圖表格 #
# 1000次模擬實驗，計算在不同樣本下，各個test的的次數，成功拒絕H0的次數(P value < 0.05 ) #
kable(PoWeR_20, "html") %>%
  kable_styling(bootstrap_options = "striped", full_width = F) %>% 
  add_header_above(c(" Number of sample" = 1, "t distribution with df=20"=6))
Number of sample
t distribution with df=20
KS.test AD.test SF.test CVM.test Lillie.test Pearson.test
n = 10 0.052 0.066 0.077 0.068 0.067 0.052
n = 50 0.068 0.079 0.106 0.076 0.069 0.068
n = 100 0.037 0.072 0.150 0.068 0.056 0.037

Normal(0,1)

# 抽取樣本數，實驗模擬1000次，計算各檢定的型二誤差機率 #
sample <- c(10,50,100)
PPP <- NULL
for(j in 1:3){
  PPP1 <- NULL
  for(i in 1:1000){
    t1 <- rnorm(sample[j],0,1)
    PPP1 <- cbind(PPP1,pvalue(t1))}
  PPP <- rbind(PPP,PPP1)}

PoWeR_30 <- apply(PPP<0.05,1,sum)/1000
PoWeR_30 <- matrix(PoWeR_30,3,6,byrow = T)
colnames(PoWeR_30) <- c("KS.test","AD.test","SF.test","CVM.test","Lillie.test","Pearson.test")
rownames(PoWeR_30) <- c("n = 10","n = 50","n = 100")
# 繪圖表格 #
# 1000次模擬實驗，計算Type 2 error的次數，成功拒絕H0的次數(P value < 0.05 ) #
kable(PoWeR_30, "html") %>%
  kable_styling(bootstrap_options = "striped", full_width = F) %>% 
  add_header_above(c(" Number of sample" = 1, "Normal (0,1)"=6))
Number of sample
Normal (0,1)
KS.test AD.test SF.test CVM.test Lillie.test Pearson.test
n = 10 0.050 0.055 0.064 0.054 0.055 0.050
n = 50 0.049 0.043 0.060 0.044 0.044 0.049
n = 100 0.048 0.048 0.049 0.052 0.044 0.048

Question 03

Write your own R programs to perform Gap test, Permutation test, and run test. Then use this program to test if the uniform random numbers generated from Minitab (or SAS, SPSS, Excel) and R are independent.

H0: This is an independent data.
H1: This is not an independent data.

從Ｒ及Excel的Uniform (0,1)生成10000萬筆隨機資料

# R
set.seed(4)
x <- runif(10000,0,1)
s <- matrix(x,ncol = 5)
kable(s) %>% kable_styling() %>% scroll_box(height = "200px")
0.5858003 0.5700942 0.0816046 0.8041510 0.4674454
0.0089458 0.8946768 0.9947983 0.4854066 0.6025949
0.2937396 0.9543947 0.5204467 0.5506201 0.1963998
0.2773750 0.8954874 0.5630114 0.1429626 0.4571380
0.8135742 0.0889351 0.8636837 0.7160434 0.1726868
0.2604278 0.2847723 0.7249578 0.6693663 0.2846362
0.7244059 0.8589519 0.4195778 0.4378635 0.8774441
0.9060922 0.2013447 0.2045223 0.2865571 0.6684647
0.9490402 0.3241199 0.3344127 0.2051451 0.1236929
0.0731445 0.5401050 0.9685382 0.9429808 0.4521745
0.7546750 0.2477951 0.3341375 0.7440445 0.0834907
0.2860006 0.2607162 0.0298998 0.6408531 0.3445729
0.1000535 0.4750125 0.2923501 0.1403950 0.7030711
0.9540688 0.6425048 0.0644463 0.1854579 0.3613593
0.4156071 0.9511689 0.1604631 0.0302305 0.7568778
0.4551024 0.0399354 0.0600415 0.0730844 0.6831844
0.9710557 0.4241846 0.8267065 0.1077859 0.7873406
0.5839880 0.5968283 0.3799005 0.4633502 0.2300417
0.9622046 0.4094558 0.7730293 0.3469409 0.8554002
0.7617024 0.9883630 0.4688608 0.7641727 0.3106805
0.7145085 0.5527471 0.0619835 0.9145523 0.6368679
0.9966129 0.7101121 0.0453856 0.8541995 0.1685104
0.5062709 0.6514963 0.7051182 0.7325002 0.6293646
0.4899432 0.7420995 0.9180690 0.2382784 0.3492051
0.6491614 0.4399088 0.9511740 0.5908027 0.0925802
0.8308064 0.3571979 0.7552401 0.3590622 0.5923435
0.4819990 0.6843168 0.6427136 0.2409987 0.2341776
0.8417462 0.8213829 0.0995630 0.4048008 0.6266874
0.5137017 0.2903821 0.3553928 0.6739971 0.6967107
0.5297775 0.4781493 0.7930540 0.8644769 0.7707295
0.5671122 0.2923825 0.8432472 0.8896607 0.4597843
0.2389489 0.6178671 0.8524523 0.7418190 0.7734486
0.8779959 0.7902776 0.2125522 0.8651776 0.6278789
0.6545220 0.7118098 0.5376848 0.8733639 0.7584641
0.4823709 0.5049059 0.2824365 0.0990720 0.8541303
0.9710298 0.8923706 0.6627202 0.5056029 0.7738381
0.4600259 0.8052235 0.8221222 0.8189084 0.1432131
0.6220565 0.5998917 0.0785730 0.9321959 0.6604426
0.3884180 0.0414823 0.6775783 0.9800260 0.0606933
0.0065927 0.7767392 0.7538228 0.8911791 0.2925328
0.9383191 0.2362328 0.1168412 0.3476032 0.0602811
0.2421711 0.4552515 0.1724259 0.6334600 0.5240131
0.5655945 0.3204748 0.0338562 0.1311921 0.8502992
0.1808991 0.0027095 0.4990582 0.3675125 0.3101432
0.9044993 0.0173521 0.3930143 0.6119740 0.3444374
0.0842913 0.6974766 0.4792338 0.8915849 0.9092571
0.9011717 0.8762575 0.3660456 0.6514626 0.6154510
0.8916115 0.2175804 0.2830181 0.0872123 0.5482808
0.7233749 0.5719707 0.8238410 0.6303855 0.6808363
0.5648099 0.3998442 0.9819282 0.6961974 0.0524000
0.3886100 0.3835307 0.3202691 0.0100274 0.3735433
0.7459289 0.1912238 0.0043386 0.4622730 0.7873403
0.8954210 0.1555778 0.4557019 0.2981048 0.7031763
0.8090767 0.1611040 0.1321982 0.5568686 0.5412488
0.8185463 0.2063833 0.1964725 0.7131293 0.0594632
0.4212000 0.9205342 0.5935186 0.2636657 0.0509218
0.1766965 0.2598682 0.1050540 0.4309326 0.2080738
0.1737866 0.9726169 0.0293268 0.6477814 0.6360684
0.8925457 0.6801529 0.9649000 0.1644722 0.5926625
0.7433341 0.0138704 0.7114975 0.0073893 0.7256524
0.5609839 0.2798266 0.1292487 0.5497285 0.9585138
0.0712338 0.5000917 0.3901010 0.5314797 0.2721367
0.8535848 0.2017655 0.1258333 0.9697406 0.6791648
0.9137110 0.7790019 0.6813997 0.4220996 0.8041856
0.2253612 0.9013658 0.5939482 0.5567293 0.0574644
0.6285601 0.4510399 0.5921234 0.9509114 0.4558010
0.0691450 0.1121876 0.4280055 0.7324646 0.6499025
0.5148540 0.9031574 0.8265878 0.7760034 0.6722316
0.8054173 0.4919271 0.2194130 0.3404867 0.1774450
0.9717895 0.3039185 0.6385384 0.7160138 0.7240594
0.3429152 0.8403735 0.7748635 0.7547099 0.3444232
0.6313634 0.2473510 0.8121583 0.2450977 0.9301418
0.4100543 0.0055159 0.4549495 0.1724148 0.2990894
0.3456941 0.9099850 0.8889945 0.8626646 0.5988580
0.8248729 0.5335462 0.0211524 0.6585161 0.6728860
0.6899005 0.2819666 0.8540851 0.2697086 0.3692808
0.3206450 0.1315159 0.4317670 0.5008032 0.6914057
0.4413297 0.8953942 0.1445119 0.7413456 0.9991377
0.2618854 0.9282858 0.7118693 0.6527987 0.0441796
0.1332045 0.1367565 0.4235067 0.8405636 0.8134832
0.9104787 0.2806165 0.4597606 0.7599409 0.6535732
0.7091679 0.9117855 0.0625651 0.4258935 0.0677557
0.5720263 0.8494691 0.3012963 0.8386789 0.1078289
0.9169682 0.5586167 0.2979471 0.3372008 0.7150386
0.9019101 0.6343287 0.1114868 0.3305393 0.6490629
0.0586220 0.2901926 0.5019482 0.1533550 0.7696756
0.0457009 0.8211197 0.6934540 0.2887509 0.4970875
0.9888085 0.5315399 0.3846532 0.4977444 0.5897383
0.2058250 0.4909442 0.7791431 0.0372737 0.4675926
0.9290560 0.9502266 0.5588727 0.2774408 0.2692785
0.1943036 0.4048742 0.6903002 0.8389700 0.6107509
0.1268009 0.7525348 0.3598623 0.8864977 0.9528540
0.5393688 0.9944917 0.7717417 0.8521102 0.1723806
0.2723371 0.3761402 0.7954990 0.6840323 0.5499520
0.3535866 0.5431205 0.6031090 0.3616487 0.3312597
0.0841049 0.9193150 0.7768232 0.6894907 0.5111104
0.7654377 0.9404167 0.7920934 0.2023641 0.8454803
0.4444860 0.7811353 0.0471133 0.1762109 0.2104052
0.0361375 0.6861349 0.2002667 0.3471496 0.5650691
0.7012891 0.3655398 0.5696570 0.4504648 0.0044335
0.2534274 0.6480416 0.4118012 0.8023511 0.4315032
0.6296312 0.2993222 0.8089402 0.2712922 0.8003153
0.2664036 0.7750032 0.6893054 0.1528213 0.6754948
0.5324459 0.6085766 0.0565364 0.6145536 0.0196848
0.4682648 0.0127904 0.8438721 0.8659620 0.3411277
0.5743969 0.9620608 0.4536139 0.3239466 0.6439015
0.6684455 0.8163858 0.7260019 0.9610442 0.4991872
0.2121137 0.1754323 0.1965188 0.7457979 0.2431923
0.9756324 0.1115895 0.3831604 0.0797697 0.7010859
0.8093828 0.5650934 0.3643944 0.5598158 0.1309380
0.2753343 0.7977886 0.6221453 0.1308882 0.5589424
0.3991237 0.1567096 0.8660354 0.9871819 0.8345207
0.2903004 0.6621076 0.3363986 0.8833318 0.3497883
0.4929932 0.3318546 0.5244160 0.7191232 0.9655189
0.7567751 0.7974442 0.5315219 0.4856904 0.3077322
0.5106975 0.5446168 0.6518966 0.8612069 0.5880637
0.4381490 0.2158349 0.6640279 0.6269400 0.2561070
0.7909141 0.3746944 0.8325431 0.9519492 0.6149660
0.9113152 0.6908660 0.4843698 0.7232863 0.2615084
0.0710741 0.9553345 0.9368189 0.7859910 0.5465989
0.1426423 0.6311856 0.0943184 0.7600007 0.3311115
0.3930728 0.7388288 0.0793942 0.0254099 0.4849566
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0.9097636 0.5324448 0.0324018 0.9277935 0.1241382
0.7944678 0.7603742 0.6046939 0.4109965 0.0681936
0.7986279 0.8125525 0.1800165 0.0402635 0.7548161
0.0377931 0.3312199 0.4915938 0.0936561 0.1069662
0.3564183 0.0618121 0.9728343 0.4068511 0.8168061
0.5410280 0.5404836 0.5180675 0.9826789 0.8442222
0.9205831 0.6518623 0.3346369 0.4671364 0.1568702
0.6275124 0.4719426 0.0261237 0.6028709 0.8136713
0.8799681 0.5300613 0.4380026 0.1485105 0.9604254
0.3308716 0.4362718 0.6622786 0.0535084 0.8632712
0.1828355 0.6807587 0.0016399 0.1384289 0.7748569
0.7158987 0.7118026 0.4014804 0.4354795 0.5239751
0.5628819 0.8633087 0.1972614 0.1888018 0.5203516
0.8990808 0.6001652 0.6095417 0.6741110 0.0473614
0.5563721 0.6877273 0.5919417 0.7083773 0.1902052
0.0304585 0.9782949 0.9682836 0.7206090 0.4370829
0.8968443 0.9384144 0.5315561 0.3806584 0.4216048
0.9178767 0.0803877 0.3865579 0.4380803 0.6514216
0.3615589 0.5605929 0.7862185 0.7048990 0.9378191
0.3821316 0.7387882 0.5881057 0.5876239 0.3519161
0.5211960 0.2323347 0.9575474 0.1970217 0.5845675
0.3337193 0.3277126 0.1253717 0.0973537 0.9760155
0.0300665 0.1539450 0.3587399 0.6819277 0.5905776
0.0082638 0.5384356 0.6562126 0.2685221 0.0780976
0.8933318 0.3451179 0.5055400 0.4812119 0.3248608
0.8600373 0.2871711 0.8800152 0.3289108 0.0011933
0.1743244 0.0040594 0.4804446 0.7552609 0.1862382
0.1965405 0.2567997 0.0551387 0.0251741 0.4510059
0.8355363 0.5815090 0.7022558 0.3303902 0.6501360
0.5688455 0.5145085 0.2064058 0.3436649 0.6023215
0.5006775 0.7217665 0.6120825 0.7502971 0.8107722
0.3382327 0.1131455 0.7474078 0.6173932 0.8735059
0.1632578 0.0036113 0.3301589 0.1459373 0.4921668
0.4628162 0.3325500 0.5467713 0.1336721 0.8353499
0.8058848 0.8823593 0.3107542 0.9387262 0.5609165
0.4034708 0.8470487 0.8020713 0.5138860 0.2116498
0.0594250 0.1060758 0.2069325 0.9116945 0.1992711
0.4340845 0.4904995 0.4414249 0.8607967 0.2767053
0.6844597 0.8975007 0.5152095 0.5583165 0.4818776
0.1843050 0.7470571 0.5495270 0.6529133 0.1218871
0.1196667 0.0713507 0.5962074 0.5738236 0.1503236
0.6309861 0.9871554 0.7337650 0.0652641 0.6082867
0.9613514 0.4118778 0.7023012 0.8673111 0.0884100
0.0071799 0.1478579 0.2703493 0.0284764 0.7963234
0.4304373 0.3108911 0.1962367 0.1233550 0.7270979
0.1762673 0.3693616 0.0734059 0.5298284 0.1928360
0.3123967 0.7703246 0.4908882 0.6863501 0.6021319
0.7612746 0.8685364 0.2559222 0.7980146 0.6050074
0.3276883 0.7940341 0.4643557 0.6098801 0.7823540
0.7934580 0.1944435 0.1756417 0.4966440 0.4550128
0.9650620 0.9376941 0.3176730 0.8869958 0.0742828
0.9652154 0.2961204 0.0426861 0.8633457 0.5759430
0.9434276 0.2076328 0.3913860 0.0709663 0.9528889
0.7233694 0.0675607 0.6202869 0.6597554 0.5165228
0.6074712 0.3310560 0.2857226 0.1542268 0.6988523
0.3080363 0.4085430 0.9996736 0.2114031 0.8083652
0.0281691 0.5445812 0.7489196 0.3078715 0.3221692
0.6169542 0.0125243 0.4528749 0.6923331 0.8356111
0.8595454 0.4474352 0.6071243 0.8271675 0.6745505
0.4020950 0.3121773 0.1259359 0.7489621 0.6449086
0.2699488 0.3560735 0.7576865 0.8155855 0.7204050
0.6715989 0.1199258 0.3276905 0.1874688 0.4975015
0.5256249 0.7574418 0.4070831 0.6246978 0.8008823
0.4047599 0.3805697 0.9482760 0.1004298 0.1080526
0.5684906 0.4898390 0.3720810 0.4612135 0.3444956
0.6135315 0.8588122 0.4157831 0.3206037 0.9003550
0.3405722 0.8928172 0.2599087 0.9826906 0.0886834
0.6343635 0.6952570 0.5965894 0.4234477 0.6158979
0.2531764 0.9985663 0.1613731 0.3709327 0.2235208
0.6556235 0.9614838 0.7868084 0.0548525 0.5199286
0.7213525 0.7154329 0.5714798 0.0994541 0.1439275
0.5338898 0.1975787 0.1303789 0.3952132 0.5852918
0.5687257 0.0543579 0.0923872 0.7986004 0.8707710
0.2417081 0.4677065 0.1046381 0.2500071 0.1039925
0.1375447 0.3651781 0.3601231 0.6608856 0.2683803
0.6511954 0.1899703 0.7707935 0.8773200 0.6271542
0.7020982 0.5632235 0.1527782 0.3534889 0.9261586
0.9053070 0.7717484 0.1825760 0.1075802 0.8914859
0.8961785 0.8280252 0.2473003 0.8886344 0.6683031
0.4689318 0.5956828 0.7459589 0.6698744 0.0740471
0.7306634 0.1293202 0.8864091 0.1134977 0.3095558
0.7226226 0.2613686 0.8272599 0.4348169 0.2852671
0.0780259 0.1015494 0.0535732 0.7188355 0.2968558
0.8308894 0.2059992 0.1984354 0.8929481 0.3831345
0.6139885 0.2606164 0.2030862 0.7048794 0.3951742 
hist(x,breaks =10,main = 'Data from R',xlab = "Data")

# Excel 
y <- read.csv('/Users/kevinpiger/Desktop/碩一下/統模/Homework01/excelrandom.csv') %>% as.vector()
y <- y$x %>% as.vector()
s <- matrix(y,ncol=5)
kable(s) %>% kable_styling() %>% scroll_box(height = "200px")
0.9110574 0.4437284 0.1669938 0.0136106 0.7423490
0.1417846 0.7438941 0.6654176 0.7534800 0.6595501
0.9734443 0.6288364 0.6739286 0.7383965 0.0589680
0.6784009 0.8537533 0.7175361 0.2292809 0.0748575
0.8831056 0.0316480 0.6290301 0.5239821 0.1299312
0.3562608 0.9073767 0.1095073 0.5664341 0.7543261
0.6754294 0.2806280 0.4884399 0.0582642 0.9595825
0.9424016 0.5145542 0.2101992 0.2465682 0.7025868
0.9440206 0.1131282 0.8182079 0.0722745 0.3764681
0.1548665 0.3449568 0.6197832 0.7178906 0.3000504
0.8412873 0.6893765 0.6967223 0.5865015 0.9466245
0.5151801 0.3506917 0.8112128 0.3315308 0.9186164
0.6322537 0.0750975 0.0538643 0.0377214 0.1857732
0.2875641 0.1644089 0.2969269 0.9838700 0.2908059
0.0899366 0.2210384 0.4504231 0.9031622 0.5753976
0.5643050 0.9919634 0.2616406 0.4472121 0.7070443
0.2737482 0.9282931 0.3929859 0.2938078 0.2936992
0.8863034 0.8218196 0.9140154 0.0280166 0.2017255
0.1019968 0.3226841 0.8571283 0.8744691 0.4000681
0.2610468 0.3521939 0.7550749 0.2027815 0.9452420
0.4141146 0.3224326 0.5437112 0.1493188 0.6823843
0.0249237 0.1243892 0.1536712 0.6011406 0.8335232
0.8919593 0.6092776 0.7525000 0.3692734 0.0241845
0.1600683 0.1293394 0.2682719 0.3774095 0.4691712
0.2674591 0.8080126 0.8464009 0.1210324 0.3604864
0.1844090 0.2676894 0.4597419 0.1907426 0.6951442
0.3613989 0.0555582 0.8828091 0.8106177 0.2886570
0.0305939 0.7660103 0.3731645 0.0515061 0.4582218
0.1914742 0.3355059 0.7764919 0.6629329 0.3330945
0.1072447 0.8475774 0.5001281 0.9136823 0.3192016
0.4619431 0.2331187 0.6531242 0.2575880 0.8210244
0.8774540 0.0253174 0.0582027 0.2822543 0.9562550
0.3693076 0.5089276 0.2135971 0.8482350 0.7784200
0.9528981 0.5453475 0.9257687 0.2851338 0.9052100
0.3576701 0.6548206 0.3938353 0.2443815 0.8649310
0.3612942 0.5700964 0.1901895 0.3202706 0.8947513
0.2721183 0.6103810 0.5149337 0.7877933 0.0848261
0.4920075 0.6728179 0.4905547 0.4418994 0.6722737
0.1698220 0.0501897 0.7531543 0.0040288 0.9048268
0.1984017 0.5382801 0.2646416 0.7122129 0.4236256
0.5379488 0.8734822 0.8317914 0.1616082 0.8751270
0.3055683 0.6148780 0.9172442 0.1489883 0.2589212
0.6862593 0.2542650 0.1228170 0.0459758 0.6887973
0.9598958 0.4324067 0.1851854 0.7157116 0.6164884
0.9690524 0.4598131 0.4106096 0.9641101 0.3203387
0.8641009 0.0781540 0.1147387 0.7987430 0.9328224
0.9442246 0.5341610 0.4137776 0.4728602 0.9464559
0.5828475 0.6441668 0.3595558 0.3612028 0.0850872
0.9186451 0.5107272 0.0548027 0.7358724 0.0611138
0.6687228 0.7915959 0.0682271 0.8080351 0.1395105
0.2246367 0.3516082 0.6926620 0.6455230 0.7537848
0.4686196 0.4789653 0.5710566 0.3053563 0.8615065
0.0893850 0.9705701 0.7476813 0.1241499 0.3392483
0.2943854 0.3717880 0.2797193 0.5879248 0.7454696
0.7351954 0.6402652 0.2424981 0.2514979 0.1081907
0.4292455 0.9377350 0.6663949 0.9257570 0.3608053
0.3291905 0.5113098 0.0990436 0.1982544 0.0541019
0.7049957 0.5833621 0.6265965 0.0608638 0.2898701
0.8627459 0.5667988 0.2070623 0.9379320 0.8460937
0.1703823 0.1876966 0.0962235 0.8237580 0.2965766
0.6146588 0.6168728 0.2291573 0.8998914 0.5624421
0.5700643 0.7816953 0.4459949 0.4745753 0.9636380
0.0709406 0.9522600 0.8367664 0.1868322 0.8633602
0.2992323 0.6331963 0.5335629 0.0879696 0.4948660
0.1977521 0.1302016 0.5913911 0.5056351 0.2124749
0.6192871 0.2991144 0.5103020 0.2097158 0.0662614
0.3588763 0.2162328 0.6453694 0.6936931 0.6557730
0.6340011 0.2243961 0.7234058 0.8996007 0.5766886
0.6570478 0.4253764 0.2816904 0.5897782 0.4046826
0.0029319 0.3003501 0.3697113 0.4025137 0.5006404
0.2757444 0.9840436 0.7383759 0.0485429 0.2624513
0.4356084 0.8213414 0.8830658 0.8603956 0.0182095
0.2699679 0.2845881 0.6868480 0.6695540 0.0478298
0.3513056 0.0722681 0.8537962 0.1933831 0.8755399
0.3932810 0.6094562 0.7523783 0.1905689 0.1984454
0.8743056 0.1304468 0.2222868 0.8922037 0.2724770
0.4549511 0.4197918 0.9747489 0.2679964 0.5215989
0.3634395 0.4401436 0.6052031 0.2163130 0.5125633
0.3281232 0.4937957 0.6480214 0.5727387 0.6507272
0.7666052 0.2236356 0.2964258 0.0185932 0.7725853
0.3331213 0.6440545 0.0280837 0.4964685 0.8406154
0.7696717 0.6231556 0.0024681 0.1460717 0.2225902
0.8730343 0.3763137 0.4811110 0.0265372 0.0731198
0.0873899 0.7043959 0.0331961 0.0111271 0.9242221
0.7627133 0.7814979 0.9273544 0.0128387 0.4010856
0.9218610 0.6354838 0.0456671 0.7806077 0.0451153
0.7177776 0.5762702 0.5270166 0.6738372 0.2533873
0.6887714 0.3726820 0.5674781 0.1812902 0.6800137
0.1801996 0.6664559 0.6046554 0.9450577 0.9906747
0.6148009 0.1248537 0.4429981 0.5852083 0.2700292
0.9585313 0.4160262 0.4692123 0.5953946 0.3814906
0.0362017 0.1522477 0.0503574 0.7964003 0.7123343
0.4426705 0.8276649 0.3565557 0.1005643 0.2028662
0.9623127 0.5637321 0.6309925 0.1850219 0.5718162
0.5903351 0.6455061 0.0914004 0.6625835 0.5149001
0.7612783 0.0216583 0.1667707 0.0404147 0.9258196
0.8040667 0.0108675 0.9145988 0.2506272 0.2494375
0.9490711 0.6506005 0.6628098 0.2906343 0.2958027
0.0386579 0.6428658 0.8444528 0.6913335 0.5555720
0.7231134 0.6461681 0.7187053 0.2428868 0.4989270
0.3666209 0.1467091 0.2799085 0.1988384 0.4658499
0.7972284 0.7405706 0.4225555 0.8762109 0.5397686
0.0175629 0.7708103 0.8897000 0.4764783 0.8903698
0.1791160 0.0083428 0.1871899 0.1713250 0.4446864
0.4030118 0.2171064 0.1001004 0.4590416 0.8437923
0.4193535 0.9077684 0.3875710 0.1121689 0.6166483
0.0747710 0.8627611 0.9065675 0.2228553 0.0082270
0.6759875 0.4251384 0.6798852 0.5288596 0.2707823
0.3217339 0.3012006 0.8309412 0.5439189 0.0383106
0.3808517 0.2789071 0.6281644 0.6455389 0.8863950
0.9743287 0.5913849 0.5591989 0.5728225 0.6411306
0.5429379 0.4055773 0.4565292 0.4280597 0.4824631
0.1572103 0.5382262 0.8870989 0.3998108 0.7567775
0.2330473 0.9685895 0.4705938 0.6197564 0.1598677
0.8258402 0.0842751 0.2707880 0.2460956 0.8968688
0.8959789 0.4113075 0.1340115 0.1284471 0.6736516
0.7180625 0.8455036 0.3304684 0.8106229 0.0616838
0.4769519 0.3790562 0.1830104 0.1396310 0.7196283
0.1305234 0.7980716 0.8562762 0.7784274 0.7925707
0.7065251 0.1885897 0.4345566 0.0296102 0.7355096
0.5678922 0.6267377 0.5927981 0.6580735 0.7100334
0.5648177 0.5809879 0.1584359 0.2413476 0.5309756
0.8918252 0.6642743 0.8314094 0.3286730 0.1069095
0.9055574 0.4579067 0.4981344 0.0075124 0.8287103
0.7029798 0.0386105 0.1443598 0.2610935 0.1344555
0.9816655 0.9262872 0.2559742 0.1978003 0.7929194
0.8524962 0.1097967 0.1589073 0.4298016 0.5955427
0.9035815 0.3525597 0.7542856 0.6752920 0.2865945
0.4945054 0.4716813 0.2787431 0.6323783 0.7939957
0.1528303 0.5483734 0.8357829 0.3820101 0.6864379
0.6184137 0.5113742 0.0026744 0.4435727 0.9615589
0.6790103 0.6658429 0.9491240 0.1269989 0.9200145
0.1255455 0.8213525 0.9263403 0.4701528 0.6845330
0.0434439 0.4710200 0.0021202 0.8577763 0.9466215
0.1610340 0.4339194 0.6334749 0.6456869 0.8669467
0.4979010 0.8838362 0.8124914 0.0589705 0.7732959
0.2225271 0.6352394 0.5431646 0.1164546 0.7844688
0.0126468 0.4681542 0.9679477 0.2532752 0.5672266
0.5539287 0.2669211 0.2974721 0.7966136 0.3777429
0.8799281 0.1435985 0.6132702 0.6850336 0.7251510
0.9522397 0.4600562 0.2317138 0.3604537 0.6131692
0.2927179 0.1637379 0.4136309 0.1445435 0.5352074
0.7102663 0.9426298 0.8943669 0.3421856 0.2303707
0.4451777 0.7793016 0.6246044 0.1125947 0.8398853
0.1008936 0.7217949 0.7258155 0.3790134 0.9528076
0.7184997 0.2070077 0.7808581 0.0785152 0.8372961
0.8249343 0.1777287 0.8827803 0.6043478 0.4359562
0.6713663 0.0858786 0.8883758 0.2734151 0.1154127
0.6534010 0.3609874 0.9312810 0.2878589 0.7414296
0.7105019 0.1159340 0.0396554 0.0444666 0.2079806
0.4048431 0.5033474 0.4880025 0.3503157 0.5299881
0.1978890 0.7602428 0.8572157 0.7565957 0.5096691
0.9196058 0.4011877 0.2247183 0.1042184 0.0090753
0.8140296 0.7610624 0.8411065 0.5982440 0.5285723
0.3946954 0.1765132 0.4766783 0.6877069 0.7138517
0.6451726 0.6581156 0.5329228 0.2906769 0.7049730
0.4165848 0.9493992 0.8328804 0.4071255 0.4814139
0.5408307 0.5523663 0.2213688 0.5574526 0.1231928
0.7421615 0.6210114 0.5448193 0.1060010 0.5009401
0.5086852 0.3385286 0.7782074 0.5586265 0.3010365
0.4716127 0.6497690 0.3315489 0.8352448 0.5200650
0.3945631 0.6673887 0.3431846 0.9595474 0.7318377
0.4214194 0.8017209 0.9037412 0.1125960 0.9958716
0.7956130 0.5233134 0.1788875 0.4003090 0.6148224
0.8670035 0.3278461 0.5618611 0.9928367 0.3207983
0.7282773 0.1098685 0.1990840 0.6064295 0.6573393
0.1569158 0.5599494 0.0050590 0.2604241 0.9018242
0.2842462 0.0693338 0.0258400 0.9475119 0.9596192
0.3254903 0.2926391 0.2926809 0.8324126 0.3202204
0.5162273 0.3845263 0.0880386 0.3587165 0.9440950
0.2320442 0.7330929 0.6648691 0.9474903 0.4052620
0.9663225 0.0929457 0.4553578 0.4697387 0.2390855
0.9819877 0.1380527 0.1984113 0.8976139 0.3106913
0.2666219 0.2515201 0.6984911 0.1968598 0.7878807
0.1146632 0.2989574 0.5402165 0.6226253 0.9107837
0.1442935 0.5776820 0.4182335 0.4638322 0.5422587
0.1410923 0.1006527 0.2500520 0.6281654 0.7415037
0.3379107 0.6694728 0.6233935 0.5751099 0.4529681
0.2649283 0.8294740 0.3738901 0.8728525 0.0351747
0.6501504 0.9686914 0.9704370 0.0327189 0.1819026
0.0772790 0.7970995 0.1347847 0.9063120 0.2374612
0.8286636 0.8508560 0.3258181 0.3865817 0.0096308
0.3494550 0.3370820 0.0240973 0.2780373 0.8653760
0.2896930 0.3363781 0.0027522 0.9721732 0.3743419
0.8705159 0.5062809 0.2566598 0.3149798 0.5651144
0.7610623 0.0633951 0.6815298 0.8658693 0.8782476
0.1745488 0.4821301 0.4720886 0.6652140 0.7080548
0.6421308 0.1608687 0.3937151 0.2525217 0.2775153
0.2924633 0.7205808 0.1693114 0.1319203 0.1988680
0.4311832 0.8022621 0.6165061 0.1839581 0.3752358
0.8964348 0.6187754 0.6178894 0.7832351 0.5889259
0.3797728 0.7575634 0.8679898 0.8318639 0.0778788
0.8415703 0.3686894 0.3052005 0.1364673 0.9087533
0.2723448 0.5626650 0.5044528 0.6056051 0.4167608
0.2983230 0.7108957 0.3387130 0.4046982 0.4984939
0.9153965 0.0247854 0.7488156 0.7627611 0.1876787
0.0688247 0.5675276 0.3430487 0.7260517 0.3159959
0.7363591 0.4369898 0.6191771 0.7516605 0.9429679
0.9876475 0.4867671 0.5093018 0.1580482 0.4612535
0.3917340 0.0952277 0.8346804 0.3158553 0.2875246
0.8737687 0.4918874 0.4736748 0.5795139 0.4265266
0.4297777 0.1522544 0.0518549 0.8907633 0.6321552
0.2745975 0.9404585 0.5255957 0.0586962 0.6330070
0.1607009 0.2863348 0.6869402 0.5077875 0.9492208
0.9007243 0.4294096 0.4039964 0.3851981 0.5543504
0.4741185 0.0873596 0.9682270 0.0241646 0.9666585
0.5099778 0.2522931 0.9908753 0.1337487 0.6301766
0.1977312 0.2903939 0.6409795 0.9139622 0.3783336
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0.9276888 0.7085929 0.3169455 0.7639303 0.1663137
0.6648985 0.3212371 0.9022653 0.3769223 0.2339018
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0.6195299 0.7529612 0.8185484 0.4829786 0.4382656
0.4396783 0.0184358 0.3427923 0.4207092 0.9305024
0.6728814 0.8511009 0.3096886 0.8590224 0.9532460
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0.9787690 0.1383980 0.5868829 0.6253932 0.2428160
0.1703741 0.0559860 0.7409828 0.9833093 0.0083302
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0.1326207 0.0429145 0.4961136 0.0295232 0.5728926
0.9553809 0.2642444 0.1806017 0.1966477 0.6065075
0.0873231 0.1549357 0.3728942 0.0586298 0.5707226
0.6400284 0.0048101 0.2332063 0.3912542 0.1355406
0.9576896 0.8435300 0.4987331 0.8097282 0.0310598
0.8899141 0.2086884 0.2075367 0.1020842 0.0224532
0.7859971 0.4267268 0.0701199 0.7299781 0.3711451
0.2527601 0.9968209 0.5046069 0.7415712 0.8356729
0.1384813 0.5696588 0.9285133 0.5878464 0.1543051
0.4553800 0.2557070 0.5236766 0.9352200 0.4066226
0.5714317 0.6671661 0.4334003 0.2428582 0.1061869
0.0525496 0.0607634 0.1583815 0.7171410 0.6831635
0.2010829 0.2498077 0.9175321 0.9884263 0.9294808
0.6008212 0.5174688 0.9621507 0.4801079 0.7835379
0.0021215 0.0987099 0.8664925 0.1730054 0.9220640
0.6561322 0.0170698 0.1387197 0.7011134 0.1300355
0.6143881 0.8920241 0.4613619 0.6121688 0.5067128
0.0205724 0.2483562 0.1090573 0.7208165 0.3226500
0.7600796 0.1230515 0.9259524 0.7632658 0.7790689
0.6586244 0.1268815 0.4816589 0.2087325 0.8102069
0.5001146 0.4968433 0.2406613 0.1671796 0.1466449
0.4258777 0.4460250 0.7943578 0.7869572 0.6601169
0.7269862 0.3421699 0.7722539 0.3891917 0.5845437
0.4574618 0.8493078 0.2717702 0.1446883 0.4267978
0.5605381 0.3162825 0.6409747 0.7754977 0.1904440
0.9643298 0.7598712 0.8617172 0.7893819 0.7923419
0.4912553 0.1552598 0.8815021 0.1409477 0.8902109
0.5274573 0.4507813 0.4060722 0.9083317 0.7747361
0.9746357 0.2818716 0.8550428 0.3316661 0.9904428
0.7027144 0.4165325 0.7037645 0.3119548 0.3721921
0.5217151 0.6623756 0.1695908 0.0235754 0.4330216
0.4654694 0.5462070 0.3123981 0.2313176 0.7932736
0.1434648 0.1005060 0.4746154 0.7544472 0.5498132
0.2130273 0.2048552 0.8617548 0.9942585 0.7096627
0.3493841 0.0015342 0.5122057 0.5021349 0.3007004
0.0985744 0.7854684 0.6418053 0.3813164 0.8722047
0.7399250 0.3667552 0.8211425 0.7853974 0.1447131
0.9202615 0.0545881 0.9419458 0.1747052 0.1931362
0.8350210 0.4617970 0.2829521 0.2710004 0.0403705
0.1979673 0.4221967 0.5763467 0.7039628 0.5077560
0.2370185 0.8605625 0.6591023 0.5033032 0.8553216
0.5699077 0.4736403 0.5326930 0.0170978 0.3897561
0.4385805 0.4728836 0.9716629 0.3620523 0.6310148
0.2218793 0.7546422 0.7382105 0.0132987 0.4654304
0.1256584 0.2720202 0.1030612 0.5118866 0.4892038
0.9406653 0.8431871 0.1491216 0.2778544 0.0485519
0.7624089 0.4463049 0.2872993 0.8988206 0.0112646
0.8064756 0.0471483 0.6390178 0.4778783 0.3234960
0.4356072 0.4214058 0.9718857 0.7006859 0.9971791
0.2497368 0.5678265 0.4821733 0.4276088 0.5891620
0.3261233 0.4596796 0.8870775 0.8205278 0.0462393
0.1535232 0.8353748 0.1119426 0.6111227 0.1442313
0.2637146 0.1448262 0.4201086 0.1395276 0.0953798
0.2516424 0.0938013 0.7645165 0.0398386 0.0478857
0.3539429 0.5183723 0.2295229 0.5676631 0.8154085
0.7177903 0.2826724 0.5915888 0.7137745 0.5714236
0.9013121 0.8744597 0.8326036 0.4077914 0.9156194
0.3519122 0.0444306 0.5679740 0.7507181 0.8159416
0.5885357 0.7452513 0.9382829 0.3190656 0.5301041
0.5201640 0.4386566 0.7213420 0.5348245 0.4594134
0.3960681 0.5011542 0.5947600 0.7952716 0.3605465
0.7172936 0.8978949 0.1319313 0.1297270 0.7044641
0.5529553 0.9198154 0.3696935 0.3219232 0.9279671
0.5194510 0.3370227 0.4380233 0.5633743 0.3429933
0.4137716 0.3408013 0.8573702 0.6320371 0.6887079
0.2590418 0.8473268 0.8213981 0.6467606 0.1132072
0.7149895 0.0215694 0.2376834 0.1055058 0.6735300
0.8290741 0.5161484 0.7441081 0.2352358 0.0181405
0.2490178 0.9061320 0.2255931 0.6083278 0.8877662
0.2414265 0.3604064 0.5435125 0.1657610 0.6868923
0.6548513 0.3507227 0.8139638 0.9453946 0.5995631
0.0857133 0.5958720 0.2898547 0.2467177 0.8566727
0.5829394 0.8203698 0.5871583 0.5844217 0.0978335
0.4626547 0.9547826 0.3688559 0.3748177 0.2871698
0.8381811 0.0303917 0.3603572 0.5608463 0.4624888
0.3094153 0.7930377 0.5240233 0.1441262 0.0499275
0.3425958 0.5850371 0.2593104 0.3294222 0.1308602
0.0082834 0.7184917 0.2301863 0.5982935 0.3680893
0.2195418 0.6893190 0.7413475 0.5192978 0.4776838
0.8382161 0.3845734 0.8271038 0.8383561 0.4324723
0.8971834 0.5243780 0.1341859 0.2506353 0.5626581
0.9620232 0.2210159 0.2622264 0.4272201 0.5940324
0.7235856 0.6141532 0.2384489 0.2888122 0.9025328
0.3030993 0.0731083 0.6099705 0.0669698 0.8688566
0.1905129 0.7304960 0.7739751 0.5609407 0.8732199
0.9510359 0.4462157 0.1999327 0.7300382 0.2063781
0.0597455 0.5465750 0.2696344 0.7521981 0.5966574
0.1426871 0.2866718 0.7446434 0.1939672 0.0216781
0.1424741 0.0925861 0.2210732 0.0060804 0.3443022
0.5616910 0.0939355 0.5779862 0.1930436 0.6865266
0.3405574 0.7746441 0.2147937 0.4832752 0.4520126
0.7487042 0.4431539 0.0371350 0.4071160 0.9763885
0.4711455 0.0884006 0.1272623 0.3984443 0.1620919
0.5425468 0.7496130 0.8973968 0.6527593 0.2777450
0.5839445 0.7448921 0.5483865 0.9252954 0.0596395
0.3547771 0.4014423 0.7311621 0.4405145 0.3607402
0.7385978 0.0406072 0.6415473 0.7275849 0.9611807
0.6126282 0.4859856 0.4856157 0.5188073 0.5639070
0.4413341 0.9599504 0.7426749 0.5948413 0.5852704
0.5028244 0.8866864 0.1362477 0.4983841 0.6402540
0.9689448 0.5382229 0.9146659 0.3413521 0.7488064
0.0560688 0.9126466 0.7891965 0.1048572 0.1891040
0.3474833 0.8521654 0.0259980 0.3358035 0.2708843
0.1525247 0.3444645 0.9478862 0.8499813 0.7526142
0.4824684 0.4143606 0.1241952 0.6357795 0.1865855
0.8470703 0.1588432 0.3488983 0.5459207 0.9421149
0.7113295 0.6778077 0.9337147 0.2896995 0.1246598
0.3154709 0.9136730 0.9424447 0.9787859 0.1579948
0.1202236 0.1017416 0.6678815 0.4548235 0.4179594
0.5973018 0.9702790 0.0848919 0.2185761 0.6436936
0.8517610 0.4794541 0.7775371 0.6080558 0.5589638
0.5463080 0.1850568 0.0658817 0.5944257 0.5050111
0.7977481 0.2503829 0.2740419 0.5125669 0.7209410
0.7526675 0.1859932 0.8227076 0.7116555 0.8546282
0.0829161 0.9883506 0.2464903 0.7942660 0.7364801
0.5703775 0.2087924 0.7619681 0.2280487 0.0213765
0.3353949 0.1746994 0.3983240 0.8138672 0.2752098
0.9816536 0.1725760 0.6322490 0.6660630 0.4503730
0.6513585 0.4847391 0.2091439 0.5214877 0.4184588
0.3824262 0.0107713 0.0812994 0.6434228 0.0378425
0.4377250 0.0330387 0.3989766 0.0069362 0.0190234
0.8444398 0.2812746 0.5994263 0.5759067 0.7265154
0.5004122 0.3825715 0.5585846 0.2642086 0.5439851
0.4279216 0.8787931 0.1309498 0.5543480 0.7568129
0.0789088 0.8751751 0.8739924 0.9264858 0.7548988
0.2202203 0.0672208 0.1896516 0.4467101 0.5833635
0.2434040 0.7800190 0.4749372 0.8562148 0.5906927
0.8904753 0.7785594 0.2691420 0.4016950 0.7725638
0.2191356 0.2473136 0.4692518 0.2878213 0.4799215
0.0118846 0.6003351 0.7154109 0.4128551 0.0405480
0.7445336 0.8322607 0.9116409 0.8561992 0.4906358
0.3766876 0.8060052 0.9488514 0.1402786 0.1155922
0.9882191 0.5302072 0.3450107 0.6626218 0.7574150
0.9977492 0.1930089 0.5955370 0.6838832 0.8730950
0.1710955 0.9013847 0.1904163 0.0244927 0.1081598
0.6017920 0.5725675 0.3261173 0.6490146 0.8418074
0.3186692 0.1427391 0.0526334 0.9878800 0.2572189
0.8729955 0.0154508 0.6100877 0.2996124 0.0785340
0.4353908 0.6811938 0.7434340 0.5852199 0.9212453
0.6126586 0.8250110 0.8950922 0.7904668 0.3699948
0.9536796 0.9603693 0.8153328 0.3757698 0.5029599
0.4925874 0.9271082 0.2984411 0.5632201 0.2469583
0.9157570 0.9070674 0.8997646 0.0395601 0.6285258
0.1283050 0.0815709 0.3434706 0.8859760 0.6330093
0.4219133 0.9618907 0.7101963 0.5981340 0.9877735
0.0966985 0.4975940 0.2699466 0.8376863 0.5094948
0.2112243 0.0623201 0.9920126 0.9937023 0.0784757
0.0468954 0.4135636 0.7565030 0.1542490 0.9407265
0.1712412 0.7632139 0.5451743 0.4633181 0.7897451
0.0512108 0.3365463 0.7446948 0.9881198 0.2457740
0.7002244 0.3329933 0.0851044 0.3292847 0.7238304
0.6706829 0.6186411 0.3497972 0.2877302 0.4179538
0.1671789 0.5011899 0.0423640 0.8811799 0.5503094
0.7749515 0.4982166 0.0116993 0.9902951 0.0500757
0.6102247 0.5258549 0.6302726 0.8894878 0.6223066
0.0461240 0.0432617 0.9915674 0.6212534 0.1075668
0.2061953 0.0991705 0.2740165 0.4061510 0.8759514
0.5244397 0.7582725 0.3959044 0.1790140 0.1154689
0.2575559 0.2859567 0.9647517 0.6883091 0.6850159
0.7418905 0.0734351 0.5813275 0.4107359 0.0614081
0.9531904 0.2242394 0.3720297 0.2379153 0.0856585
0.2712682 0.7912161 0.7037240 0.6426340 0.6628245
0.2046730 0.9693058 0.4888767 0.7493875 0.0910539
0.9391307 0.1227258 0.5501988 0.9558415 0.3428719
0.9697325 0.6531892 0.1905069 0.8282284 0.6480056
0.2934175 0.1516561 0.8492305 0.0355251 0.0295622
0.4685174 0.8841297 0.0161973 0.0697663 0.8525836
0.3711683 0.5677538 0.2286560 0.5623969 0.3721919
0.2253009 0.2383669 0.0215580 0.2051767 0.4289127
0.6315501 0.2319973 0.3254667 0.4047735 0.7359348
0.4619828 0.1786923 0.1194617 0.0274867 0.8570190
0.5443546 0.2820645 0.7923365 0.9695401 0.9183434
0.9673109 0.6582069 0.7989085 0.0605166 0.5979008
0.5936129 0.4826867 0.2548997 0.1020668 0.9186319
0.8527116 0.5150548 0.0987541 0.4358866 0.4462896
0.5231286 0.5264986 0.7597531 0.9461062 0.7891599
0.2223037 0.8618241 0.1704299 0.2066543 0.4097575
0.2578444 0.6775521 0.4154059 0.2392127 0.7939856
0.5900758 0.6184598 0.7273204 0.4483779 0.5167309
0.4037359 0.4536173 0.0738757 0.8877963 0.6961608
0.5889977 0.9465892 0.6288299 0.1928828 0.3742081
0.2845954 0.3246028 0.7448807 0.7806421 0.3150724
0.1941559 0.5991910 0.2094771 0.2519632 0.4218454
0.1789687 0.6024567 0.6817397 0.7461626 0.9559504
0.9277347 0.4890347 0.9986428 0.7546030 0.6578379
0.4369463 0.2064920 0.1895146 0.6118033 0.2819482
0.7570991 0.5104160 0.1718871 0.5784550 0.7039625
0.5641635 0.5623994 0.9073007 0.0925352 0.4969873
0.8964886 0.2469226 0.0023001 0.2393195 0.8661027
0.2831550 0.0273354 0.6583901 0.2427922 0.5885573
0.9853801 0.4262753 0.5622221 0.6087862 0.8826423
0.2834751 0.4093438 0.2670090 0.8699450 0.5687260
0.3657507 0.8409609 0.6198865 0.1655893 0.5777910
0.1720765 0.0303666 0.4330336 0.0586226 0.9338153
0.0897842 0.3718695 0.9956635 0.2704855 0.6340518
0.0023195 0.0105060 0.1170145 0.0501194 0.5084864
0.9841862 0.5736831 0.6632946 0.3563575 0.1304417
0.2172669 0.8911197 0.9931120 0.2997924 0.3343513
0.6053613 0.0484593 0.2329642 0.6103307 0.4421105
0.3075782 0.4547249 0.4292715 0.8286422 0.5504933
0.4675926 0.5620389 0.7669115 0.9897472 0.1401454
0.8286720 0.1882797 0.4819488 0.6814122 0.4241236
0.4902983 0.4163246 0.1128109 0.4943025 0.2457185 
hist(y,breaks = 10,main = 'Data from EXCEL',xlab = "Data")

Run test

function: run(w,c),w為資料，須介於0,1之間，c為顯著水準；以0.5作為分界點，大於0.5定義為S成功，小於等於0.5定義為F失敗，\(n_1\)為S的數量，\(n_2\)為F的數量，Runs為總Run的個數，當\(n_1\) > 10且 \(n_2\)>10時，Runs為近似常態分配N(\(\frac{2n_1n_2}{n_1+n_2}+1\),\(\frac{2n_1n_2 (2n_1n_2-n_1-n_2)}{((n_1+n_2 )^2 (n_1+n_2-1)}\)) Rejection region:{ \(|Z_0| ≥ |Z_{α/2}|\) }，\(Z_0\)為Runs標準化的檢定統計量，α=0.05。(\(n_2\)以總個數length(rx)減去n1的個數表示) function: run(w,c); w為欲檢定資料, c為顯著水準。

run =  function(w,c){
  rx <- w
  rx[rx > 0.5] = "S"
  rx[rx <= 0.5] = "F"
  # 總ＲＵＮ數 #
  RUNS <- sum(rx[-1] != rx[-(length(rx))]) + 1
  # test #
  n1 <- sum(rx == "S")
  muR <- (2 * n1 *(length(rx)- n1))/ length(rx) + 1
  sigma_R <- 2 * n1 *(length(rx)- n1) *( 2*n1*(length(rx) - n1) - length(rx))/(length(rx)^2 * (length(rx) -1))
  z2 <-( RUNS - muR ) / sqrt(sigma_R)
  return(print(ifelse(abs(z2) >= qnorm(1-c/2,0,1), paste("Z = ",round(abs(z2),digits = 3),' >= ',qnorm(1-c/2,0,1),"Reject Ho "),paste("Z= ",round(abs(z2),digits = 3),' < ',qnorm(1-c/2,0,1),"Do not reject Ho "))))}
# 第一個位置請放資料(vector),第二個位置放顯著水準 #
run(x,0.05)
## [1] "Z=  0.22  <  1.95996398454005 Do not reject Ho "
run(y,0.05)
## [1] "Z=  1.253  <  1.95996398454005 Do not reject Ho "

Gap test

function: gap(w,c,d1,d2); w為欲檢定資料, c為顯著水準, 選取介於 0, 1間的兩個數0< d1 < d2 < 1，alpha為d1值, beta為d2值 考慮介於α與β間的亂數，介於α與β間的亂數為 1其餘為 0，間隔應與幾何分配有關，間隔個數P(K = k) = (β − α)×(1−(β −α ))k , k = 0,1,… 最後在代入卡方檢定

gap <- function(w,c,d1,d2){
sample <- w
alpha <- min(d1); beta <- max(d2)
fgap <- (sample <= beta & alpha <= sample)*1
fgap_p <- which(fgap == 1)
tmp <- fgap_p[-1]-fgap_p[-length(fgap_p)] - 1
tmp1 <- tmp %>% table() %>% as.data.frame();tmp1
names(tmp1) <- c('Gap','Freq')
p <- chisq.test(as.integer(tmp1$Freq), p =  dgeom(as.integer(tmp1$Gap),prob = beta - alpha),rescale.p = T)
return(print(ifelse(p$p.value >= c,paste("p_value",round(p$p.value,digits = 3),' > ',c,"Do not reject H0"),paste("p_value",round(p$p.value,digits = 3),'<=',c, 'Reject H0'))))}

# 第一個位置請放資料(vector),第二個位置放顯著水準 #
gap(x,0.05,0.2,0.8)
## [1] "p_value 0.713  >  0.05 Do not reject H0"
gap(y,0.05,0.2,0.8)
## [1] "p_value 0.888  >  0.05 Do not reject H0"

Permutation test

function: Permutation.test(w,c); W為欲檢定資料, c為顯著水準。

permutation.test <- function(W,c){
  W <- matrix(W , nrow = 4 ,byrow = F )
  W1 <- apply(W , 2 , rank)
  W2 <- W1[1,]*1000+W1[2,]*100+W1[3,]*10+W1[4,];W2
  p <- chisq.test(table(W2))
  return(print(ifelse(p$p.value >= c,paste("p_value",p$p.value,'>',c,"Do not reject H0"),
                      paste("p_value",p$p.value,'<=',c, 'Reject H0'))))
}

permutation.test(x,0.05)
## [1] "p_value 0.944744354483845 > 0.05 Do not reject H0"
permutation.test(y,0.05)
## [1] "p_value 0.388354963998629 > 0.05 Do not reject H0"

Conclusion Both data of R and Excel are random.

Question 04

(a)

\(\sum\limits_{i=1}^{12}U_i-6\) can be used to approximate N(0,1) distribution, where \(U_i\)’s are random sample from U(0,1). Based on α = 0.05, compare the results of the Chi-square test and the Kolmogorov-Smirnov test, and see if there are any differences.
H0: This is an approximate N(0,1) distribution.
H1: This is not an approximate N(0,1) distribution. 
# 抽10000組 x, 每組都是12個U抽自U(0,1),去形成1000個近似常態資料樣本 #
# 再用ks test 以及 chisq test去檢測是否近似常態資料, alpha =0.05 #
N <- 10000
x <- NULL
for(i in 1:N){
  x[i] <- runif(12) %>% sum() - 6}
# 繪製長條圖 #
h <- hist(x,main = "Simulation of x", xlab = 'Data')

# ks test #
p1 <- ks.test(x,"pnorm")
print(ifelse(p1$p.value >= 0.05,paste('p value =',round(p1$p.value,digits = 3),"> 0.05","Do not reject H0."),paste('p value =',round(p1$p.value,digits = 3),"<= 0.05","Reject H0.")))
## [1] "p value = 0.882 > 0.05 Do not reject H0."
# Chisq test #
p2 <- chisq.test(h$counts,dnorm(h$counts))
print(ifelse(p2$p.value >= 0.05,paste('p value =',round(p2$p.value,digits = 3),"> 0.05","Do not reject H0."),paste('p value =',round(p2$p.value,digits = 3),"<= 0.05","Reject H0.")))
## [1] "p value = 0.216 > 0.05 Do not reject H0."

Conclusion: ks.test以及Chisq.test都不拒絕H0。

#4-b
#Design two tests of independence (which are not the same as you saw in class) and apply them on the random sample that you generate.
#(1) idea by Lin, YC
y <- rank(x)
plot(1:N,y,ylab = "Rank",xlab = 'Oeder of x')
abline(v = 5000,col = 'red')
abline(h = 5000,col = 'red')

r1 <- length(which(y[1:5000]<5000))
r2 <- length(which(y[1:5000]>=5000))
r3 <- length(which(y[5001:10000]<5000))
r4 <- length(which(y[5001:10000]>=5000))
R <- c(r1,r2,r3,r4)
R0 <-sum((R-2500)^2/2500)
ifelse(R0 >= qchisq(0.95,3),paste('chisq value',R0,'>=', qchisq(0.95,3),'Reject H0'),paste('chisq value',R0,'<', qchisq(0.95,3),"Do not reject H0"))
## [1] "chisq value 0.9608 < 7.81472790325118 Do not reject H0"
#(2) rank + run.test
y <- y / 10000
run(y,0.05)
## [1] "Z=  0.02  <  1.95996398454005 Do not reject Ho "

Question 05

Use the bisection, false positions, and secant methods to find the roots, and check your answers with the functions in R. You need to specify the starting points, convergence criterion, and number of iterations.

the question of (a) is \(f(x)=e-\frac{1}{3.5+x}\)

the question of (b) is \(f(x)=\frac{e^x}{\sqrt{1+x^2}}-0.5\)

the question of (c) is the eigenvalue of matrix A \(\begin{bmatrix} 2&3&4\\ 3&4&5\\ 4&5&6 \end{bmatrix}\)

Method

Bisection Method

bisection.method<-function(f,a,b){
fa <- f(a) ; fb <- f(b)
c <- (a+b)/2 ; fc <- f(c)
na=c();nb=c();nc=c();nfa=c();nfb=c();nfc=c()
if((fa*fb)>0){
  print("Wrong Input")
}
i <- 0
W1 <- c(a,b,fa,fb,c,fc)
while (abs(b-a) > 0.00001){
  i<-i+1
  fa <- f(a) ; fb <- f(b) ; fc <- f(c)
  if((fa*fc)<0){ 
    a = a ; fa = fa
    b = c ; fb = fc
    na = c(na,a) ; nb = c(nb,b)
    nfa = c(nfa,fa) ; nfb = c(nfb,fb)
  }
  else{ 
    a = c ; fa = fc
    b = b ; fb = fb
    na = c(na,a) ; nb = c(nb,b) ; nfa = c(nfa,fa) ; nfb = c(nfb,fb)
  }
  c = (a+b)/2 ; fc <- f(c)
  nc = c(nc,c) ; nfc = c(nfc,fc)
}
W2 <- c(na,nb,nfa,nfb,nc,nfc)
W2 <- matrix(data = W2,ncol = 6,byrow = F)
W2 <- rbind(W1,W2)
W <- data.frame("a"=W2[,1],"b"=W2[,2],"func.a"=W2[,3],"func.b"=W2[,4],"c"=W2[,5],"func.c"=W2[,6])
b=b
print('covegence criterion : |b-a|<1e-6')
print(paste('number of iterations : ',i))
print(paste('approximation root :', b))
print(knitr::kable(W,"rst",digits = 7))
kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" Bisection Method" = 6))%>% kable_styling() %>% scroll_box(height = "200px")
}

False Position Method

false.position.method <- function(f,a,b){
fa <- f(a) ; fb <- f(b)
c <- a-(fa*(b-a)/(fb-fa)) ; fc <- f(c)
e=abs(c-b)
na=c();nb=c();nc=c();ni=c();nfa=c();nfb=c();nfc=c()
if((fa*fb)>0){
  print("Wrong Input")
}
i=0
W1<-c(a,b,c,fa,fb,fc)
while(e>0.00001){
  i <- i+1
  fa <- f(a) ; fb <- f(b)
  c <- b-(fb*(b-a)/(fb-fa)) ;  fc <- f(c)
  if(sign(fc) == sign(fa)){
    e <- abs(c-a)
    a <- c ; fa <- fc
    b <- b ; fb <- fb
    na = c(na,a) ; nb = c(nb,b)
    nfa=c(nfa,fa) ; nfb = c(nfb,fb)
  }
  else{
    e <- abs(c-b)
    a <- a ; fa <- fa
    b <- c ; fb <- fc
    na = c(na,a) ; nb = c(nb,b)
    nfa = c(nfa,fa) ; nfb = c(nfb,fb)
  }
  nfc = c(nfc,fc) ; nc = c(nc,c)
}
W2<-c(na,nb,nc,nfa,nfb,nfc)
W2<-matrix(data=c(na,nb,nc,nfa,nfb,nfc),ncol=6,byrow = F)
W<-rbind(W1,W2)
W<-data.frame("a"=W2[,1],"b"=W2[,2],"c"=W2[,3],"func.a"=W2[,4],"func.b"=W2[,5],"func.c"=W2[,6])
b=b
print('covegence criterion : |Cn-Cn-1|<1e-5')
print(paste('number of iterations : ',i))
print(paste('approximation root :', c))
print(knitr::kable(W,"rst",digits = 7))
kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" False Position Method" = 6))%>% kable_styling() %>% scroll_box(height = "200px")
}

Secant Method

secant.method<-function(f,a,b){
  fa <- f(a) ; fb <- f(b)
  c <- b-((b-a)/(fb-fa))*fb;fc <- f(c)
  d = abs(b-a)
  W1 <- c(a,b,fa,fb,c,fc)
  na=c();nb=c();nc=c();nfa=c();nfb=c();nfc=c()
  i=0
  while(d>0.000001){
    i <- i+1
    fa <- f(a)
    fb <- f(b)
    na = c(na,a) ;  nb = c(nb,b) ; nfa = c(nfa,fa); nfb = c(nfb,fb)
    c <- b-((b-a)/(fb-fa))*fb
    a = b ; b = c
    c2 <- b-((b-a)/(fb-fa))*fb
    nc = c(nc,c2) ; fc <- f(c2) ; nfc = c(nfc,fc)
    d = abs(b-a)
  }
  W2 <- c(na,nb,nc,nfa,nfb,nfc)
  W2 <- matrix(data=c(na,nb,nc,nfa,nfb,nfc),ncol=6,byrow = F)
  W2 <- rbind(W1,W2)
  W <- data.frame("a"=W2[,1],"b"=W2[,2],"c"=W2[,3],"func.a"=W2[,4],"func.b"=W2[,5],"func.c"=W2[,6])
  b = b
  print('covegence criterion : |Cn-Cn-1|<1e-5')
  print(paste('number of iterations : ',i))
  print(paste('approximation root :', b))
  print(knitr::kable(W,"rst",digits = 7))
  kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" Secant Method" = 6))%>% kable_styling() %>% scroll_box(height = "200px")
}

Solve problem

(a)

The question of (a)
\(f(x)=e-\frac{1}{3.5+x}\) 

f <- function(x){
  return(exp(1)-(1/(3.5+x)))
}
curve(f,xlim = c(-5,0))
abline(h=0)

當\(x=-3.5\)，其函數值不存在


當\(x<-3.5\)，其函數值皆為正


我們選取?\(a=-3.2 ,b=-3\)作為起始點


#### R_uniroot

# We choose start point a=-3.2 b=-3.0
uniroot(f,c(-3.2,-3))
## $root
## [1] -3.132115
## 
## $f.root
## [1] 0.00004459005
## 
## $iter
## [1] 4
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 0.00006103516

Bisection Method

# We choose start point a=-3.2 b=-3.0
bisection.method(f,-3.2,-3)
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  15"
## [1] "approximation root : -3.13211669921875"
## 
## 
## =========  =========  ==========  =========  =========  ==========
##         a          b      func.a     func.b          c      func.c
## =========  =========  ==========  =========  =========  ==========
## -3.200000  -3.000000  -0.6150515  0.7182818  -3.100000   0.2182818
## -3.200000  -3.100000  -0.6150515  0.2182818  -3.150000  -0.1388610
## -3.150000  -3.100000  -0.1388610  0.2182818  -3.125000   0.0516152
## -3.150000  -3.125000  -0.1388610  0.0516152  -3.137500  -0.0403389
## -3.137500  -3.125000  -0.0403389  0.0516152  -3.131250   0.0064174
## -3.137500  -3.131250  -0.0403389  0.0064174  -3.134375  -0.0167609
## -3.134375  -3.131250  -0.0167609  0.0064174  -3.132812  -0.0051224
## -3.132812  -3.131250  -0.0051224  0.0064174  -3.132031   0.0006597
## -3.132812  -3.132031  -0.0051224  0.0006597  -3.132422  -0.0022283
## -3.132422  -3.132031  -0.0022283  0.0006597  -3.132227  -0.0007835
## -3.132227  -3.132031  -0.0007835  0.0006597  -3.132129  -0.0000617
## -3.132129  -3.132031  -0.0000617  0.0006597  -3.132080   0.0002991
## -3.132129  -3.132080  -0.0000617  0.0002991  -3.132105   0.0001187
## -3.132129  -3.132105  -0.0000617  0.0001187  -3.132117   0.0000285
## -3.132129  -3.132117  -0.0000617  0.0000285  -3.132123  -0.0000166
## -3.132123  -3.132117  -0.0000166  0.0000285  -3.132120   0.0000060
## =========  =========  ==========  =========  =========  ==========
Bisection Method
a b func.a func.b c func.c
-3.200000 -3.000000 -0.6150515 0.7182818 -3.100000 0.2182818
-3.200000 -3.100000 -0.6150515 0.2182818 -3.150000 -0.1388610
-3.150000 -3.100000 -0.1388610 0.2182818 -3.125000 0.0516152
-3.150000 -3.125000 -0.1388610 0.0516152 -3.137500 -0.0403389
-3.137500 -3.125000 -0.0403389 0.0516152 -3.131250 0.0064174
-3.137500 -3.131250 -0.0403389 0.0064174 -3.134375 -0.0167609
-3.134375 -3.131250 -0.0167609 0.0064174 -3.132812 -0.0051224
-3.132812 -3.131250 -0.0051224 0.0064174 -3.132031 0.0006597
-3.132812 -3.132031 -0.0051224 0.0006597 -3.132422 -0.0022283
-3.132422 -3.132031 -0.0022283 0.0006597 -3.132227 -0.0007835
-3.132227 -3.132031 -0.0007835 0.0006597 -3.132129 -0.0000617
-3.132129 -3.132031 -0.0000617 0.0006597 -3.132080 0.0002991
-3.132129 -3.132080 -0.0000617 0.0002991 -3.132105 0.0001187
-3.132129 -3.132105 -0.0000617 0.0001187 -3.132117 0.0000285
-3.132129 -3.132117 -0.0000617 0.0000285 -3.132123 -0.0000166
-3.132123 -3.132117 -0.0000166 0.0000285 -3.132120 0.0000060

False Position Method

# We choose start point a=-3.2 b=-3.0
false.position.method(f,-3.2,-3)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  7"
## [1] "approximation root : -3.13211959677426"
## 
## 
## ====  =========  =========  ==========  =========  =========
##    a          b          c      func.a     func.b     func.c
## ====  =========  =========  ==========  =========  =========
## -3.2  -3.107742  -3.107742  -0.6150515  0.1689375  0.1689375
## -3.2  -3.127622  -3.127622  -0.6150515  0.0328357  0.0328357
## -3.2  -3.131291  -3.131291  -0.6150515  0.0061190  0.0061190
## -3.2  -3.131967  -3.131967  -0.6150515  0.0011311  0.0011311
## -3.2  -3.132092  -3.132092  -0.6150515  0.0002088  0.0002088
## -3.2  -3.132115  -3.132115  -0.6150515  0.0000385  0.0000385
## -3.2  -3.132120  -3.132120  -0.6150515  0.0000071  0.0000071
## ====  =========  =========  ==========  =========  =========
False Position Method
a b c func.a func.b func.c
-3.2 -3.107742 -3.107742 -0.6150515 0.1689375 0.1689375
-3.2 -3.127622 -3.127622 -0.6150515 0.0328357 0.0328357
-3.2 -3.131291 -3.131291 -0.6150515 0.0061190 0.0061190
-3.2 -3.131967 -3.131967 -0.6150515 0.0011311 0.0011311
-3.2 -3.132092 -3.132092 -0.6150515 0.0002088 0.0002088
-3.2 -3.132115 -3.132115 -0.6150515 0.0000385 0.0000385
-3.2 -3.132120 -3.132120 -0.6150515 0.0000071 0.0000071

Secant Method

# We choose start point a=-3.2 b=-3.0
secant.method(f,-3.2,-3)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  6"
## [1] "approximation root : -3.13212055882774"
## 
## 
## =========  =========  ==========  ==========  ==========  ==========
##         a          b           c      func.a      func.b      func.c
## =========  =========  ==========  ==========  ==========  ==========
## -3.200000  -3.000000  -0.6150515   0.7182818  -3.1077423   0.1689375
## -3.200000  -3.000000  -3.0497003  -0.6150515   0.7182818   0.4975387
## -3.000000  -3.107742  -3.1510652   0.7182818   0.1689375  -0.1475832
## -3.107742  -3.140876  -3.1341706   0.1689375  -0.0662701  -0.0152331
## -3.140876  -3.131540  -3.1320724  -0.0662701   0.0042803   0.0003559
## -3.131540  -3.132107  -3.1321209   0.0042803   0.0001020  -0.0000027
## -3.132107  -3.132121  -3.1321206   0.0001020  -0.0000002   0.0000000
## =========  =========  ==========  ==========  ==========  ==========
Secant Method
a b c func.a func.b func.c
-3.200000 -3.000000 -0.6150515 0.7182818 -3.1077423 0.1689375
-3.200000 -3.000000 -3.0497003 -0.6150515 0.7182818 0.4975387
-3.000000 -3.107742 -3.1510652 0.7182818 0.1689375 -0.1475832
-3.107742 -3.140876 -3.1341706 0.1689375 -0.0662701 -0.0152331
-3.140876 -3.131540 -3.1320724 -0.0662701 0.0042803 0.0003559
-3.131540 -3.132107 -3.1321209 0.0042803 0.0001020 -0.0000027
-3.132107 -3.132121 -3.1321206 0.0001020 -0.0000002 0.0000000

Conclusion

(b)

The question of (b)
\({f(x)=\frac{e^x}{\sqrt{1+x^2}}-0.5}\) 

f<-function(x){
  return((exp(-x))/sqrt(1+x^2)-0.5)
}
curve(f,xlim = c(-0,1))
abline(h=0)

由圖形觀察出其根介於(0,1)之間/center>

我們選取?\(a=0 ,b=1\)作為起始點


R_uniroot

# We choose start point a=0 b=1
uniroot(f,c(0,1))
## $root
## [1] 0.5577421
## 
## $f.root
## [1] -0.000009543937
## 
## $iter
## [1] 5
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 0.00006103516

Bisection Method

# We choose start point a=0 b=1
bisection.method(f,0,1)
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  17"
## [1] "approximation root : 0.557731628417969"
## 
## 
## =========  =========  =========  ==========  =========  ==========
##         a          b     func.a      func.b          c      func.c
## =========  =========  =========  ==========  =========  ==========
## 0.0000000  1.0000000  0.5000000  -0.2398700  0.5000000   0.0424975
## 0.5000000  1.0000000  0.0424975  -0.2398700  0.7500000  -0.1221068
## 0.5000000  0.7500000  0.0424975  -0.1221068  0.6250000  -0.0460992
## 0.5000000  0.6250000  0.0424975  -0.0460992  0.5625000  -0.0033912
## 0.5000000  0.5625000  0.0424975  -0.0033912  0.5312500   0.0191569
## 0.5312500  0.5625000  0.0191569  -0.0033912  0.5468750   0.0077835
## 0.5468750  0.5625000  0.0077835  -0.0033912  0.5546875   0.0021712
## 0.5546875  0.5625000  0.0021712  -0.0033912  0.5585938  -0.0006162
## 0.5546875  0.5585938  0.0021712  -0.0006162  0.5566406   0.0007760
## 0.5566406  0.5585938  0.0007760  -0.0006162  0.5576172   0.0000795
## 0.5576172  0.5585938  0.0000795  -0.0006162  0.5581055  -0.0002685
## 0.5576172  0.5581055  0.0000795  -0.0002685  0.5578613  -0.0000945
## 0.5576172  0.5578613  0.0000795  -0.0000945  0.5577393  -0.0000075
## 0.5576172  0.5577393  0.0000795  -0.0000075  0.5576782   0.0000360
## 0.5576782  0.5577393  0.0000360  -0.0000075  0.5577087   0.0000142
## 0.5577087  0.5577393  0.0000142  -0.0000075  0.5577240   0.0000033
## 0.5577240  0.5577393  0.0000033  -0.0000075  0.5577316  -0.0000021
## 0.5577240  0.5577316  0.0000033  -0.0000021  0.5577278   0.0000006
## =========  =========  =========  ==========  =========  ==========
Bisection Method
a b func.a func.b c func.c
0.0000000 1.0000000 0.5000000 -0.2398700 0.5000000 0.0424975
0.5000000 1.0000000 0.0424975 -0.2398700 0.7500000 -0.1221068
0.5000000 0.7500000 0.0424975 -0.1221068 0.6250000 -0.0460992
0.5000000 0.6250000 0.0424975 -0.0460992 0.5625000 -0.0033912
0.5000000 0.5625000 0.0424975 -0.0033912 0.5312500 0.0191569
0.5312500 0.5625000 0.0191569 -0.0033912 0.5468750 0.0077835
0.5468750 0.5625000 0.0077835 -0.0033912 0.5546875 0.0021712
0.5546875 0.5625000 0.0021712 -0.0033912 0.5585938 -0.0006162
0.5546875 0.5585938 0.0021712 -0.0006162 0.5566406 0.0007760
0.5566406 0.5585938 0.0007760 -0.0006162 0.5576172 0.0000795
0.5576172 0.5585938 0.0000795 -0.0006162 0.5581055 -0.0002685
0.5576172 0.5581055 0.0000795 -0.0002685 0.5578613 -0.0000945
0.5576172 0.5578613 0.0000795 -0.0000945 0.5577393 -0.0000075
0.5576172 0.5577393 0.0000795 -0.0000075 0.5576782 0.0000360
0.5576782 0.5577393 0.0000360 -0.0000075 0.5577087 0.0000142
0.5577087 0.5577393 0.0000142 -0.0000075 0.5577240 0.0000033
0.5577240 0.5577393 0.0000033 -0.0000075 0.5577316 -0.0000021
0.5577240 0.5577316 0.0000033 -0.0000021 0.5577278 0.0000006

False Position Method

# We choose start point a=0 b=1
false.position.method(f,0,1)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  8"
## [1] "approximation root : 0.557730709781261"
## 
## 
## ===  =========  =========  ======  ==========  ==========
##   a          b          c  func.a      func.b      func.c
## ===  =========  =========  ======  ==========  ==========
##   0  0.6757944  0.6757944     0.5  -0.0784769  -0.0784769
##   0  0.5841154  0.5841154     0.5  -0.0185218  -0.0185218
##   0  0.5632506  0.5632506     0.5  -0.0039230  -0.0039230
##   0  0.5588657  0.5588657     0.5  -0.0008098  -0.0008098
##   0  0.5579620  0.5579620     0.5  -0.0001663  -0.0001663
##   0  0.5577765  0.5577765     0.5  -0.0000341  -0.0000341
##   0  0.5577385  0.5577385     0.5  -0.0000070  -0.0000070
##   0  0.5577307  0.5577307     0.5  -0.0000014  -0.0000014
## ===  =========  =========  ======  ==========  ==========
False Position Method
a b c func.a func.b func.c
0 0.6757944 0.6757944 0.5 -0.0784769 -0.0784769
0 0.5841154 0.5841154 0.5 -0.0185218 -0.0185218
0 0.5632506 0.5632506 0.5 -0.0039230 -0.0039230
0 0.5588657 0.5588657 0.5 -0.0008098 -0.0008098
0 0.5579620 0.5579620 0.5 -0.0001663 -0.0001663
0 0.5577765 0.5577765 0.5 -0.0000341 -0.0000341
0 0.5577385 0.5577385 0.5 -0.0000070 -0.0000070
0 0.5577307 0.5577307 0.5 -0.0000014 -0.0000014

Secant Method

# We choose start point a=0 b=1
secant.method(f,0,1)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  6"
## [1] "approximation root : 0.557728698983827"
## 
## 
## =========  =========  =========  ==========  ==========  ==========
##         a          b          c      func.a      func.b      func.c
## =========  =========  =========  ==========  ==========  ==========
## 0.0000000  1.0000000  0.5000000  -0.2398700   0.6757944  -0.0784769
## 0.0000000  1.0000000  0.7809037   0.5000000  -0.2398700  -0.1390305
## 1.0000000  0.6757944  0.4414967  -0.2398700  -0.0784769   0.0882895
## 0.6757944  0.5181506  0.5491327  -0.0784769   0.0288448   0.0061565
## 0.5181506  0.5605205  0.5579664   0.0288448  -0.0019865  -0.0001694
## 0.5605205  0.5577905  0.5577272  -0.0019865  -0.0000440   0.0000011
## 0.5577905  0.5577286  0.5577287  -0.0000440   0.0000001   0.0000000
## =========  =========  =========  ==========  ==========  ==========
Secant Method
a b c func.a func.b func.c
0.0000000 1.0000000 0.5000000 -0.2398700 0.6757944 -0.0784769
0.0000000 1.0000000 0.7809037 0.5000000 -0.2398700 -0.1390305
1.0000000 0.6757944 0.4414967 -0.2398700 -0.0784769 0.0882895
0.6757944 0.5181506 0.5491327 -0.0784769 0.0288448 0.0061565
0.5181506 0.5605205 0.5579664 0.0288448 -0.0019865 -0.0001694
0.5605205 0.5577905 0.5577272 -0.0019865 -0.0000440 0.0000011
0.5577905 0.5577286 0.5577287 -0.0000440 0.0000001 0.0000000

Conclusion

(c)

The question of (c)
the eigenvalue of A=\(\begin{bmatrix} 2&3&4\\ 3&4&5\\ 4&5&6 \end{bmatrix}\) 

A<-c(2,3,4,3,4,5,4,5,6)
A<-matrix(A,ncol=3,byrow=T)
eigen(A)
## eigen() decomposition
## $values
## [1] 12.480740698407867483866 -0.000000000000000721645
## [3] -0.480740698407859934349
## 
## $vectors
##           [,1]       [,2]       [,3]
## [1,] 0.4303622  0.4082483  0.8050601
## [2,] 0.5665422 -0.8164966  0.1111905
## [3,] 0.7027221  0.4082483 -0.5826791
f <- function(x){
  return((2-x)*(4-x)*(6-x)+3*5*4+4*3*5-(2-x)*5*5-4*(4-x)*4-3*3*(6-x))
}
f(0)
## [1] 0
curve(f,xlim = c(-5,15))
abline(h=0)

因為函數為三次式，我們將圖形更進一步分割 


curve(f,xlim = c(-1,1))
abline(h=0)

curve(f,xlim = c(12,13))
abline(h=0)

我們選取三組起始點來計算三根?


\(a=-1 ,b=-0.2\)作為第一組


\(a=-0.2,b=1\)作為第二組


\(a=12 ,b=13\)作為第三組


R_uniroot

#第一組
# We choose start point a=-1 b=-0.2
uniroot(f,c(-1,-0.2))
## $root
## [1] -0.480742
## 
## $f.root
## [1] 0.000008279401
## 
## $iter
## [1] 8
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 0.00006103516
#第二組
# We choose start point a=-0.2 b=1
uniroot(f,c(-0.2,1))
## $root
## [1] -0.0000005977228
## 
## $f.root
## [1] -0.000003586333
## 
## $iter
## [1] 9
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 0.00006103516
#第三組
# We choose start point a=12 b=13
uniroot(f,c(12,13))
## $root
## [1] 12.48074
## 
## $f.root
## [1] 0.0004214783
## 
## $iter
## [1] 4
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 0.00006103516

Bisection Method

第一組(-1,-0.2)
bisection.method(f,-1,-0.2)
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  17"
## [1] "approximation root : -0.4807373046875"
## 
## 
## ==========  ==========  =========  ==========  ==========  ==========
##          a           b     func.a      func.b           c      func.c
## ==========  ==========  =========  ==========  ==========  ==========
## -1.0000000  -0.2000000  7.0000000  -0.7120000  -0.6000000   0.9360000
## -0.6000000  -0.2000000  0.9360000  -0.7120000  -0.4000000  -0.4160000
## -0.6000000  -0.4000000  0.9360000  -0.4160000  -0.5000000   0.1250000
## -0.5000000  -0.4000000  0.1250000  -0.4160000  -0.4500000  -0.1788750
## -0.5000000  -0.4500000  0.1250000  -0.1788750  -0.4750000  -0.0353281
## -0.5000000  -0.4750000  0.1250000  -0.0353281  -0.4875000   0.0427324
## -0.4875000  -0.4750000  0.0427324  -0.0353281  -0.4812500   0.0031770
## -0.4812500  -0.4750000  0.0031770  -0.0353281  -0.4781250  -0.0162068
## -0.4812500  -0.4781250  0.0031770  -0.0162068  -0.4796875  -0.0065477
## -0.4812500  -0.4796875  0.0031770  -0.0065477  -0.4804687  -0.0016935
## -0.4812500  -0.4804687  0.0031770  -0.0016935  -0.4808594   0.0007397
## -0.4808594  -0.4804687  0.0007397  -0.0016935  -0.4806641  -0.0004774
## -0.4808594  -0.4806641  0.0007397  -0.0004774  -0.4807617   0.0001310
## -0.4807617  -0.4806641  0.0001310  -0.0004774  -0.4807129  -0.0001733
## -0.4807617  -0.4807129  0.0001310  -0.0001733  -0.4807373  -0.0000211
## -0.4807617  -0.4807373  0.0001310  -0.0000211  -0.4807495   0.0000549
## -0.4807495  -0.4807373  0.0000549  -0.0000211  -0.4807434   0.0000169
## -0.4807434  -0.4807373  0.0000169  -0.0000211  -0.4807404  -0.0000021
## ==========  ==========  =========  ==========  ==========  ==========
Bisection Method
a b func.a func.b c func.c
-1.0000000 -0.2000000 7.0000000 -0.7120000 -0.6000000 0.9360000
-0.6000000 -0.2000000 0.9360000 -0.7120000 -0.4000000 -0.4160000
-0.6000000 -0.4000000 0.9360000 -0.4160000 -0.5000000 0.1250000
-0.5000000 -0.4000000 0.1250000 -0.4160000 -0.4500000 -0.1788750
-0.5000000 -0.4500000 0.1250000 -0.1788750 -0.4750000 -0.0353281
-0.5000000 -0.4750000 0.1250000 -0.0353281 -0.4875000 0.0427324
-0.4875000 -0.4750000 0.0427324 -0.0353281 -0.4812500 0.0031770
-0.4812500 -0.4750000 0.0031770 -0.0353281 -0.4781250 -0.0162068
-0.4812500 -0.4781250 0.0031770 -0.0162068 -0.4796875 -0.0065477
-0.4812500 -0.4796875 0.0031770 -0.0065477 -0.4804687 -0.0016935
-0.4812500 -0.4804687 0.0031770 -0.0016935 -0.4808594 0.0007397
-0.4808594 -0.4804687 0.0007397 -0.0016935 -0.4806641 -0.0004774
-0.4808594 -0.4806641 0.0007397 -0.0004774 -0.4807617 0.0001310
-0.4807617 -0.4806641 0.0001310 -0.0004774 -0.4807129 -0.0001733
-0.4807617 -0.4807129 0.0001310 -0.0001733 -0.4807373 -0.0000211
-0.4807617 -0.4807373 0.0001310 -0.0000211 -0.4807495 0.0000549
-0.4807495 -0.4807373 0.0000549 -0.0000211 -0.4807434 0.0000169
-0.4807434 -0.4807373 0.0000169 -0.0000211 -0.4807404 -0.0000021
第二組(-0.2,0)
bisection.method(f,-0.2,1)
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  17"
## [1] "approximation root : 0.000006103515625"
## 
## 
## ==========  =========  ==========  ==========  ==========  ==========
##          a          b      func.a      func.b           c      func.c
## ==========  =========  ==========  ==========  ==========  ==========
## -0.2000000  1.0000000  -0.7120000  17.0000000   0.4000000   4.2560000
## -0.2000000  0.4000000  -0.7120000   4.2560000   0.1000000   0.7190000
## -0.2000000  0.1000000  -0.7120000   0.7190000  -0.0500000  -0.2698750
## -0.0500000  0.1000000  -0.2698750   0.7190000   0.0250000   0.1574844
## -0.0500000  0.0250000  -0.2698750   0.1574844  -0.0125000  -0.0731230
## -0.0125000  0.0250000  -0.0731230   0.1574844   0.0062500   0.0379685
## -0.0125000  0.0062500  -0.0731230   0.0379685  -0.0031250  -0.0186328
## -0.0031250  0.0062500  -0.0186328   0.0379685   0.0015625   0.0094043
## -0.0031250  0.0015625  -0.0186328   0.0094043  -0.0007812  -0.0046802
## -0.0007812  0.0015625  -0.0046802   0.0094043   0.0003906   0.0023456
## -0.0007812  0.0003906  -0.0046802   0.0023456  -0.0001953  -0.0011714
## -0.0001953  0.0003906  -0.0011714   0.0023456   0.0000977   0.0005861
## -0.0001953  0.0000977  -0.0011714   0.0005861  -0.0000488  -0.0002929
## -0.0000488  0.0000977  -0.0002929   0.0005861   0.0000244   0.0001465
## -0.0000488  0.0000244  -0.0002929   0.0001465  -0.0000122  -0.0000732
## -0.0000122  0.0000244  -0.0000732   0.0001465   0.0000061   0.0000366
## -0.0000122  0.0000061  -0.0000732   0.0000366  -0.0000031  -0.0000183
## -0.0000031  0.0000061  -0.0000183   0.0000366   0.0000015   0.0000092
## ==========  =========  ==========  ==========  ==========  ==========
Bisection Method
a b func.a func.b c func.c
-0.2000000 1.0000000 -0.7120000 17.0000000 0.4000000 4.2560000
-0.2000000 0.4000000 -0.7120000 4.2560000 0.1000000 0.7190000
-0.2000000 0.1000000 -0.7120000 0.7190000 -0.0500000 -0.2698750
-0.0500000 0.1000000 -0.2698750 0.7190000 0.0250000 0.1574844
-0.0500000 0.0250000 -0.2698750 0.1574844 -0.0125000 -0.0731230
-0.0125000 0.0250000 -0.0731230 0.1574844 0.0062500 0.0379685
-0.0125000 0.0062500 -0.0731230 0.0379685 -0.0031250 -0.0186328
-0.0031250 0.0062500 -0.0186328 0.0379685 0.0015625 0.0094043
-0.0031250 0.0015625 -0.0186328 0.0094043 -0.0007812 -0.0046802
-0.0007812 0.0015625 -0.0046802 0.0094043 0.0003906 0.0023456
-0.0007812 0.0003906 -0.0046802 0.0023456 -0.0001953 -0.0011714
-0.0001953 0.0003906 -0.0011714 0.0023456 0.0000977 0.0005861
-0.0001953 0.0000977 -0.0011714 0.0005861 -0.0000488 -0.0002929
-0.0000488 0.0000977 -0.0002929 0.0005861 0.0000244 0.0001465
-0.0000488 0.0000244 -0.0002929 0.0001465 -0.0000122 -0.0000732
-0.0000122 0.0000244 -0.0000732 0.0001465 0.0000061 0.0000366
-0.0000122 0.0000061 -0.0000732 0.0000366 -0.0000031 -0.0000183
-0.0000031 0.0000061 -0.0000183 0.0000366 0.0000015 0.0000092
第三組(12,13)
bisection.method(f,12,13)
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  17"
## [1] "approximation root : 12.4807434082031"
## 
## 
## ========  ========  ==========  ===========  ========  ==========
##        a         b      func.a       func.b         c      func.c
## ========  ========  ==========  ===========  ========  ==========
## 12.00000  13.00000  72.0000000  -91.0000000  12.50000  -3.1250000
## 12.00000  12.50000  72.0000000   -3.1250000  12.25000  35.9843750
## 12.25000  12.50000  35.9843750   -3.1250000  12.37500  16.8222656
## 12.37500  12.50000  16.8222656   -3.1250000  12.43750   6.9475098
## 12.43750  12.50000   6.9475098   -3.1250000  12.46875   1.9360657
## 12.46875  12.50000   1.9360657   -3.1250000  12.48438  -0.5882530
## 12.46875  12.48438   1.9360657   -0.5882530  12.47656   0.6754584
## 12.47656  12.48438   0.6754584   -0.5882530  12.48047   0.0439909
## 12.48047  12.48438   0.0439909   -0.5882530  12.48242  -0.2720340
## 12.48047  12.48242   0.0439909   -0.2720340  12.48145  -0.1139973
## 12.48047  12.48145   0.0439909   -0.1139973  12.48096  -0.0349971
## 12.48047  12.48096   0.0439909   -0.0349971  12.48071   0.0044984
## 12.48071  12.48096   0.0044984   -0.0349971  12.48084  -0.0152490
## 12.48071  12.48084   0.0044984   -0.0152490  12.48077  -0.0053752
## 12.48071  12.48077   0.0044984   -0.0053752  12.48074  -0.0004384
## 12.48071  12.48074   0.0044984   -0.0004384  12.48073   0.0020300
## 12.48073  12.48074   0.0020300   -0.0004384  12.48074   0.0007958
## 12.48074  12.48074   0.0007958   -0.0004384  12.48074   0.0001787
## ========  ========  ==========  ===========  ========  ==========
Bisection Method
a b func.a func.b c func.c
12.00000 13.00000 72.0000000 -91.0000000 12.50000 -3.1250000
12.00000 12.50000 72.0000000 -3.1250000 12.25000 35.9843750
12.25000 12.50000 35.9843750 -3.1250000 12.37500 16.8222656
12.37500 12.50000 16.8222656 -3.1250000 12.43750 6.9475098
12.43750 12.50000 6.9475098 -3.1250000 12.46875 1.9360657
12.46875 12.50000 1.9360657 -3.1250000 12.48438 -0.5882530
12.46875 12.48438 1.9360657 -0.5882530 12.47656 0.6754584
12.47656 12.48438 0.6754584 -0.5882530 12.48047 0.0439909
12.48047 12.48438 0.0439909 -0.5882530 12.48242 -0.2720340
12.48047 12.48242 0.0439909 -0.2720340 12.48145 -0.1139973
12.48047 12.48145 0.0439909 -0.1139973 12.48096 -0.0349971
12.48047 12.48096 0.0439909 -0.0349971 12.48071 0.0044984
12.48071 12.48096 0.0044984 -0.0349971 12.48084 -0.0152490
12.48071 12.48084 0.0044984 -0.0152490 12.48077 -0.0053752
12.48071 12.48077 0.0044984 -0.0053752 12.48074 -0.0004384
12.48071 12.48074 0.0044984 -0.0004384 12.48073 0.0020300
12.48073 12.48074 0.0020300 -0.0004384 12.48074 0.0007958
12.48074 12.48074 0.0007958 -0.0004384 12.48074 0.0001787

False Position Method

第一組(-1,-0.2)
# We choose start point a=-1 b=-0.2
false.position.method(f,-1,-0.2)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  18"
## [1] "approximation root : -0.480731127708486"
## 
## 
## ===  ==========  ==========  ======  ==========  ==========
##   a           b           c  func.a      func.b      func.c
## ===  ==========  ==========  ======  ==========  ==========
##  -1  -0.2738589  -0.2738589       7  -0.7226300  -0.7226300
##  -1  -0.3418061  -0.3418061       7  -0.6089259  -0.6089259
##  -1  -0.3944800  -0.3944800       7  -0.4381196  -0.4381196
##  -1  -0.4301463  -0.4301463       7  -0.2809796  -0.2809796
##  -1  -0.4521375  -0.4521375       7  -0.1672557  -0.1672557
##  -1  -0.4649224  -0.4649224       7  -0.0952059  -0.0952059
##  -1  -0.4721023  -0.4721023       7  -0.0528245  -0.0528245
##  -1  -0.4760561  -0.4760561       7  -0.0288951  -0.0288951
##  -1  -0.4782100  -0.4782100       7  -0.0156828  -0.0156828
##  -1  -0.4793764  -0.4793764       7  -0.0084759  -0.0084759
##  -1  -0.4800061  -0.4800061       7  -0.0045703  -0.0045703
##  -1  -0.4803453  -0.4803453       7  -0.0024614  -0.0024614
##  -1  -0.4805280  -0.4805280       7  -0.0013247  -0.0013247
##  -1  -0.4806263  -0.4806263       7  -0.0007127  -0.0007127
##  -1  -0.4806792  -0.4806792       7  -0.0003834  -0.0003834
##  -1  -0.4807076  -0.4807076       7  -0.0002062  -0.0002062
##  -1  -0.4807229  -0.4807229       7  -0.0001109  -0.0001109
##  -1  -0.4807311  -0.4807311       7  -0.0000596  -0.0000596
## ===  ==========  ==========  ======  ==========  ==========
False Position Method
a b c func.a func.b func.c
-1 -0.2738589 -0.2738589 7 -0.7226300 -0.7226300
-1 -0.3418061 -0.3418061 7 -0.6089259 -0.6089259
-1 -0.3944800 -0.3944800 7 -0.4381196 -0.4381196
-1 -0.4301463 -0.4301463 7 -0.2809796 -0.2809796
-1 -0.4521375 -0.4521375 7 -0.1672557 -0.1672557
-1 -0.4649224 -0.4649224 7 -0.0952059 -0.0952059
-1 -0.4721023 -0.4721023 7 -0.0528245 -0.0528245
-1 -0.4760561 -0.4760561 7 -0.0288951 -0.0288951
-1 -0.4782100 -0.4782100 7 -0.0156828 -0.0156828
-1 -0.4793764 -0.4793764 7 -0.0084759 -0.0084759
-1 -0.4800061 -0.4800061 7 -0.0045703 -0.0045703
-1 -0.4803453 -0.4803453 7 -0.0024614 -0.0024614
-1 -0.4805280 -0.4805280 7 -0.0013247 -0.0013247
-1 -0.4806263 -0.4806263 7 -0.0007127 -0.0007127
-1 -0.4806792 -0.4806792 7 -0.0003834 -0.0003834
-1 -0.4807076 -0.4807076 7 -0.0002062 -0.0002062
-1 -0.4807229 -0.4807229 7 -0.0001109 -0.0001109
-1 -0.4807311 -0.4807311 7 -0.0000596 -0.0000596
第二組(-0.2,1)
# We choose start point a=-0.2 b=-1
false.position.method(f,-0.2,-1)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  18"
## [1] "approximation root : -0.480731127708486"
## 
## 
## ==========  ===  ==========  ==========  ======  ==========
##          a    b           c      func.a  func.b      func.c
## ==========  ===  ==========  ==========  ======  ==========
## -0.2738589   -1  -0.2738589  -0.7226300       7  -0.7226300
## -0.3418061   -1  -0.3418061  -0.6089259       7  -0.6089259
## -0.3944800   -1  -0.3944800  -0.4381196       7  -0.4381196
## -0.4301463   -1  -0.4301463  -0.2809796       7  -0.2809796
## -0.4521375   -1  -0.4521375  -0.1672557       7  -0.1672557
## -0.4649224   -1  -0.4649224  -0.0952059       7  -0.0952059
## -0.4721023   -1  -0.4721023  -0.0528245       7  -0.0528245
## -0.4760561   -1  -0.4760561  -0.0288951       7  -0.0288951
## -0.4782100   -1  -0.4782100  -0.0156828       7  -0.0156828
## -0.4793764   -1  -0.4793764  -0.0084759       7  -0.0084759
## -0.4800061   -1  -0.4800061  -0.0045703       7  -0.0045703
## -0.4803453   -1  -0.4803453  -0.0024614       7  -0.0024614
## -0.4805280   -1  -0.4805280  -0.0013247       7  -0.0013247
## -0.4806263   -1  -0.4806263  -0.0007127       7  -0.0007127
## -0.4806792   -1  -0.4806792  -0.0003834       7  -0.0003834
## -0.4807076   -1  -0.4807076  -0.0002062       7  -0.0002062
## -0.4807229   -1  -0.4807229  -0.0001109       7  -0.0001109
## -0.4807311   -1  -0.4807311  -0.0000596       7  -0.0000596
## ==========  ===  ==========  ==========  ======  ==========
False Position Method
a b c func.a func.b func.c
-0.2738589 -1 -0.2738589 -0.7226300 7 -0.7226300
-0.3418061 -1 -0.3418061 -0.6089259 7 -0.6089259
-0.3944800 -1 -0.3944800 -0.4381196 7 -0.4381196
-0.4301463 -1 -0.4301463 -0.2809796 7 -0.2809796
-0.4521375 -1 -0.4521375 -0.1672557 7 -0.1672557
-0.4649224 -1 -0.4649224 -0.0952059 7 -0.0952059
-0.4721023 -1 -0.4721023 -0.0528245 7 -0.0528245
-0.4760561 -1 -0.4760561 -0.0288951 7 -0.0288951
-0.4782100 -1 -0.4782100 -0.0156828 7 -0.0156828
-0.4793764 -1 -0.4793764 -0.0084759 7 -0.0084759
-0.4800061 -1 -0.4800061 -0.0045703 7 -0.0045703
-0.4803453 -1 -0.4803453 -0.0024614 7 -0.0024614
-0.4805280 -1 -0.4805280 -0.0013247 7 -0.0013247
-0.4806263 -1 -0.4806263 -0.0007127 7 -0.0007127
-0.4806792 -1 -0.4806792 -0.0003834 7 -0.0003834
-0.4807076 -1 -0.4807076 -0.0002062 7 -0.0002062
-0.4807229 -1 -0.4807229 -0.0001109 7 -0.0001109
-0.4807311 -1 -0.4807311 -0.0000596 7 -0.0000596
第三組(12,13)
# We choose start point a=12 b=12
false.position.method(f,12,13)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  6"
## [1] "approximation root : 12.4807405928187"
## 
## 
## ========  ===  ========  =========  ======  =========
##        a    b         c     func.a  func.b     func.c
## ========  ===  ========  =========  ======  =========
## 12.44172   13  12.44172  6.2740086     -91  6.2740086
## 12.47773   13  12.47773  0.4874456     -91  0.4874456
## 12.48051   13  12.48051  0.0375246     -91  0.0375246
## 12.48072   13  12.48072  0.0028867     -91  0.0028867
## 12.48074   13  12.48074  0.0002221     -91  0.0002221
## 12.48074   13  12.48074  0.0000171     -91  0.0000171
## ========  ===  ========  =========  ======  =========
False Position Method
a b c func.a func.b func.c
12.44172 13 12.44172 6.2740086 -91 6.2740086
12.47773 13 12.47773 0.4874456 -91 0.4874456
12.48051 13 12.48051 0.0375246 -91 0.0375246
12.48072 13 12.48072 0.0028867 -91 0.0028867
12.48074 13 12.48074 0.0002221 -91 0.0002221
12.48074 13 12.48074 0.0000171 -91 0.0000171

Secant Method

第一組(-1,-0.2)
# We choose start point a=-1 b=-0.2
secant.method(f,-1,-0.2)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  20"
## [1] "approximation root : -0.480740698409208"
## 
## 
## ===========  ===========  ==============  ============  ============  =======================
##           a            b               c        func.a        func.b                   func.c
## ===========  ===========  ==============  ============  ============  =======================
##  -1.0000000   -0.2000000       7.0000000    -0.7120000    -0.2738589               -0.7226300
##  -1.0000000   -0.2000000      -0.2670400     7.0000000    -0.7120000               -0.7274729
##  -0.2000000   -0.2738589    -336.5808456    -0.7120000    -0.7226300         39487542.2213939
##  -0.2738589    4.7471086       4.7283455    -0.7226300   191.9268478              190.9442797
##   4.7471086   -0.2550252      -0.2360635   191.9268478    -0.7331107               -0.7345144
##  -0.2550252   -0.2359911    5292.5871632    -0.7331107    -0.7345022    -147917024040.9768677
##  -0.2359911  -10.2826619     -10.2794476    -0.7345022  2294.3197520             2292.5266817
## -10.2826619   -0.2392064      -0.2424214  2294.3197520    -0.7349147               -0.7350641
##  -0.2392064   -0.2424225  -77762.7859485    -0.7349147    -0.7350641  470308087225702.5000000
##  -0.2424225   15.5734410      15.5885048    -0.7350641  -773.2333753             -778.4809426
##  15.5734410   -0.2574719      -0.2724924  -773.2333753    -0.7322618               -0.7236959
##  -0.2574719   -0.2724782    -108.9311142    -0.7322618    -0.7237068          1434313.5205628
##  -0.2724782   -1.5419319      -1.5043034    -0.7237068    22.9450835               21.5334559
##  -1.5419319   -0.3112935      -0.3454451    22.9450835    -0.6747517               -0.5994600
##  -0.3112935   -0.3464494      -2.6754120    -0.6747517    -0.5967868               88.9916239
##  -0.3464494   -0.6155519      -0.5539748    -0.5967868     1.0867730                0.5288168
##  -0.6155519   -0.4418406      -0.4663161     1.0867730    -0.2221087               -0.0870874
##  -0.4418406   -0.4713183      -0.4852201    -0.2221087    -0.0575196                0.0281816
##  -0.4713183   -0.4816199      -0.4808007    -0.0575196     0.0054891                0.0003738
##  -0.4816199   -0.4807225      -0.4807403     0.0054891    -0.0001134               -0.0000025
##  -0.4807225   -0.4807407      -0.4807407    -0.0001134    -0.0000002                0.0000000
## ===========  ===========  ==============  ============  ============  =======================
Secant Method
a b c func.a func.b func.c
-1.0000000 -0.2000000 7.0000000 -0.7120000 -0.2738589 -0.7226300
-1.0000000 -0.2000000 -0.2670400 7.0000000 -0.7120000 -0.7274729
-0.2000000 -0.2738589 -336.5808456 -0.7120000 -0.7226300 39487542.2213939
-0.2738589 4.7471086 4.7283455 -0.7226300 191.9268478 190.9442797
4.7471086 -0.2550252 -0.2360635 191.9268478 -0.7331107 -0.7345144
-0.2550252 -0.2359911 5292.5871632 -0.7331107 -0.7345022 -147917024040.9768677
-0.2359911 -10.2826619 -10.2794476 -0.7345022 2294.3197520 2292.5266817
-10.2826619 -0.2392064 -0.2424214 2294.3197520 -0.7349147 -0.7350641
-0.2392064 -0.2424225 -77762.7859485 -0.7349147 -0.7350641 470308087225702.5000000
-0.2424225 15.5734410 15.5885048 -0.7350641 -773.2333753 -778.4809426
15.5734410 -0.2574719 -0.2724924 -773.2333753 -0.7322618 -0.7236959
-0.2574719 -0.2724782 -108.9311142 -0.7322618 -0.7237068 1434313.5205628
-0.2724782 -1.5419319 -1.5043034 -0.7237068 22.9450835 21.5334559
-1.5419319 -0.3112935 -0.3454451 22.9450835 -0.6747517 -0.5994600
-0.3112935 -0.3464494 -2.6754120 -0.6747517 -0.5967868 88.9916239
-0.3464494 -0.6155519 -0.5539748 -0.5967868 1.0867730 0.5288168
-0.6155519 -0.4418406 -0.4663161 1.0867730 -0.2221087 -0.0870874
-0.4418406 -0.4713183 -0.4852201 -0.2221087 -0.0575196 0.0281816
-0.4713183 -0.4816199 -0.4808007 -0.0575196 0.0054891 0.0003738
-0.4816199 -0.4807225 -0.4807403 0.0054891 -0.0001134 -0.0000025
-0.4807225 -0.4807407 -0.4807407 -0.0001134 -0.0000002 0.0000000
第二組(-0.2,0)
# We choose start point a=-0.2 b=1
secant.method(f,-0.2,1)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  8"
## [1] "approximation root : -0.0000000000680793734079034"
## 
## 
## ==========  ==========  ==========  ==========  ==========  ==========
##          a           b           c      func.a      func.b      func.c
## ==========  ==========  ==========  ==========  ==========  ==========
## -0.2000000   1.0000000  -0.7120000  17.0000000  -0.1517615  -0.6306951
## -0.2000000   1.0000000   0.9537006  -0.7120000  17.0000000  15.7693093
##  1.0000000  -0.1517615  -0.1120339  17.0000000  -0.6306951  -0.5201782
## -0.1517615  -0.1105601   0.8955375  -0.6306951  -0.5153266  14.2788648
## -0.1105601   0.0734781   0.0274730  -0.5153266   0.5052603   0.1738747
##  0.0734781  -0.0176334  -0.0048949   0.5052603  -0.1020635  -0.0290816
## -0.0176334  -0.0023217   0.0004638  -0.1020635  -0.0138654   0.0027856
## -0.0023217   0.0000854   0.0000027  -0.0138654   0.0005127   0.0000160
##  0.0000854  -0.0000004   0.0000000   0.0005127  -0.0000024   0.0000000
## ==========  ==========  ==========  ==========  ==========  ==========
Secant Method
a b c func.a func.b func.c
-0.2000000 1.0000000 -0.7120000 17.0000000 -0.1517615 -0.6306951
-0.2000000 1.0000000 0.9537006 -0.7120000 17.0000000 15.7693093
1.0000000 -0.1517615 -0.1120339 17.0000000 -0.6306951 -0.5201782
-0.1517615 -0.1105601 0.8955375 -0.6306951 -0.5153266 14.2788648
-0.1105601 0.0734781 0.0274730 -0.5153266 0.5052603 0.1738747
0.0734781 -0.0176334 -0.0048949 0.5052603 -0.1020635 -0.0290816
-0.0176334 -0.0023217 0.0004638 -0.1020635 -0.0138654 0.0027856
-0.0023217 0.0000854 0.0000027 -0.0138654 0.0005127 0.0000160
0.0000854 -0.0000004 0.0000000 0.0005127 -0.0000024 0.0000000
第三組(12,13)
# We choose start point a=12 b=13
secant.method(f,12,13)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  5"
## [1] "approximation root : 12.4807406984078"
## 
## 
## ========  ========  ========  ===========  ===========  ===========
##        a         b         c       func.a       func.b       func.c
## ========  ========  ========  ===========  ===========  ===========
## 12.00000  13.00000  72.00000  -91.0000000   12.4417178    6.2740086
## 12.00000  13.00000  12.75340   72.0000000  -91.0000000  -46.0189560
## 13.00000  12.44172  12.47540  -91.0000000    6.2740086    0.8626558
## 12.44172  12.47773  12.48101    6.2740086    0.4874456   -0.0443439
## 12.47773  12.48076  12.48074    0.4874456   -0.0030080   -0.0000170
## 12.48076  12.48074  12.48074   -0.0030080    0.0000014    0.0000000
## ========  ========  ========  ===========  ===========  ===========
Secant Method
a b c func.a func.b func.c
12.00000 13.00000 72.00000 -91.0000000 12.4417178 6.2740086
12.00000 13.00000 12.75340 72.0000000 -91.0000000 -46.0189560
13.00000 12.44172 12.47540 -91.0000000 6.2740086 0.8626558
12.44172 12.47773 12.48101 6.2740086 0.4874456 -0.0443439
12.47773 12.48076 12.48074 0.4874456 -0.0030080 -0.0000170
12.48076 12.48074 12.48074 -0.0030080 0.0000014 0.0000000

Conclusion

Question 06

Consider a multinomial observation X =(\(x_1,x_2,x_3,x_4\))with class probabilities given by (\(p_1,p_2,p_3,p_4\))=(\(\frac{2+θ}{4}\),\(\frac{1-θ}{4}\),\(\frac{1-θ}{4}\),\(\frac{θ}{4}\)), where 0<\(\theta\) <1. The sample size is n= \(\sum{x_i}\) and the parameter \(\theta\) is to be estimated from the observedfrequencies (1997, 906, 904, 32), i.e., sample size 3839. Use the secant, Ridder’s (or Brent’s), and Newton-Raphosn methods to find the MLE (via l’(\(\theta\))). Youmay choose your own starting points and convergence criterion (preferred \(10^{-6}\) or smaller)

Define f.o.c likelihood function

f = function(x){(1997/(2+x)-1810/(1-x)+32/x)/4}
curve(f,xlim = c(0,0.1))
abline(h=0, col = "red")

Newton-Raphosn Method

#Newton
Newton <- function(f,S,tol){
i=1
N_table <- matrix(c(0,S,f(S)),1,3)
colnames(N_table) <- c('Iterative','x','f(x)')
tr = tol
while(tr>=tol){
  ss <- N_table[i,2] - f(N_table[i,2])/grad(f,N_table[i,2])
  tr <- abs(ss-N_table[i,2])
  i = i+1
  N_table <- rbind(N_table,c(i-1,ss,f(ss)))
}
  print("Newton's Method ")
  print('covegence criterion : tolerence <1e-6')
  print(paste('number of iterations : ',i))
  print(paste('approximation root :', ss))
  print(knitr::kable(N_table,"rst",digits = 7))
  kable(N_table, "html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c("Newton-Raphosn Method"= 3 ))
}

Ridder method by our hand

#Ridders method function
ridders.method<-function(f,a,b){
  fa <- f(a) ; fb <- f(b) ; 
  c <- (a+b)/2 ; fc <- f(c)
  e_Q <- (fc+sign(fb)*(fc^2-fa*fb)^(1/2)/fb)
  d <- c-(c-a)*sign(fa-fb)*(fc/(fc^2-fa*fb)^(1/2))
  fd <- f(d)
  na=c();nb=c();nd=c();nfa=c();nfb=c();nfd=c()
  if((fa*fb)>0){
    print("Wrong Input")
  }
  i <- 0
  W1 <- c(a,b,fa,fb,d,fd)
  while (abs(b-a) > 0.000001){
    i<-i+1
    fa <- f(a) ; fb <- f(b) ; fd <- f(d)
    
    if((fa*fd)<0){ 
      a = a ; fa = fa
      b = d ; fb = fd
      na = c(na,a) ; nb = c(nb,b)
      nfa = c(nfa,fa) ; nfb = c(nfb,fb)
    }
    else{ 
      a = d ; fa = fd
      b = b ; fb = fb
      na = c(na,a) ; nb = c(nb,b) ; nfa = c(nfa,fa) ; nfb = c(nfb,fb)
    }
    c = (a+b)/2 ; fc <- f(c)
    d <- c-(c-a)*sign(fa-fb)*(fc/(fc^2-fa*fb)^(1/2)) ; fd <- f(d)
    nd = c(nd,d) ; nfd = c(nfd,fd)
  }
  W2 <- c(na,nb,nfa,nfb,nd,nfd)
  W2 <- matrix(data = W2,ncol = 6,byrow = F)
  W2 <- rbind(W1,W2)
  W <- data.frame("a"=W2[,1],"b"=W2[,2],"func.a"=W2[,3],"func.b"=W2[,4],"d"=W2[,5],"func.d"=W2[,6])
  d=b
  print("Ridder's Method ")
  print('covegence criterion : |b-a|<1e-6')
  print(paste('number of iterations : ',i))
  print(paste('approximation root :', d))
  print(knitr::kable(W,"rst",digits = 7))
  kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" Ridder Method" = 6))
}
ridders.method(f,0.02,0.05)
## [1] "Ridder's Method "
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  51"
## [1] "approximation root : 0.0357129053949557"
## 
## 
## =========  =========  ===========  ===========  =========  ===========
##         a          b       func.a       func.b          d       func.d
## =========  =========  ===========  ===========  =========  ===========
## 0.0200000  0.0500000  185.4187715  -72.7792041  0.0343561    9.6653759
## 0.0343561  0.0500000    9.6653759  -72.7792041  0.0486078  -67.3340615
## 0.0343561  0.0486078    9.6653759  -67.3340615  0.0472218  -61.6463651
## 0.0343561  0.0472218    9.6653759  -61.6463651  0.0458414  -55.6935156
## 0.0343561  0.0458414    9.6653759  -55.6935156  0.0444663  -49.4499989
## 0.0343561  0.0444663    9.6653759  -49.4499989  0.0430959  -42.8868806
## 0.0343561  0.0430959    9.6653759  -42.8868806  0.0417296  -35.9711896
## 0.0343561  0.0417296    9.6653759  -35.9711896  0.0403668  -28.6651600
## 0.0343561  0.0403668    9.6653759  -28.6651600  0.0390069  -20.9252915
## 0.0343561  0.0390069    9.6653759  -20.9252915  0.0376491  -12.7011749
## 0.0343561  0.0376491    9.6653759  -12.7011749  0.0362926   -3.9340090
## 0.0343561  0.0362926    9.6653759   -3.9340090  0.0349365    5.4452895
## 0.0349365  0.0362926    5.4452895   -3.9340090  0.0355168    1.3520540
## 0.0355168  0.0362926    1.3520540   -3.9340090  0.0360970   -2.6210252
## 0.0355168  0.0360970    1.3520540   -2.6210252  0.0359015   -1.2951049
## 0.0355168  0.0359015    1.3520540   -1.2951049  0.0357059    0.0439843
## 0.0357059  0.0359015    0.0439843   -1.2951049  0.0358951   -1.2515407
## 0.0357059  0.0358951    0.0439843   -1.2515407  0.0358887   -1.2079624
## 0.0357059  0.0358887    0.0439843   -1.2079624  0.0358823   -1.1643700
## 0.0357059  0.0358823    0.0439843   -1.1643700  0.0358759   -1.1207636
## 0.0357059  0.0358759    0.0439843   -1.1207636  0.0358695   -1.0771430
## 0.0357059  0.0358695    0.0439843   -1.0771430  0.0358631   -1.0335083
## 0.0357059  0.0358631    0.0439843   -1.0335083  0.0358567   -0.9898595
## 0.0357059  0.0358567    0.0439843   -0.9898595  0.0358503   -0.9461966
## 0.0357059  0.0358503    0.0439843   -0.9461966  0.0358439   -0.9025196
## 0.0357059  0.0358439    0.0439843   -0.9025196  0.0358375   -0.8588283
## 0.0357059  0.0358375    0.0439843   -0.8588283  0.0358311   -0.8151230
## 0.0357059  0.0358311    0.0439843   -0.8151230  0.0358247   -0.7714035
## 0.0357059  0.0358247    0.0439843   -0.7714035  0.0358184   -0.7276698
## 0.0357059  0.0358184    0.0439843   -0.7276698  0.0358120   -0.6839219
## 0.0357059  0.0358120    0.0439843   -0.6839219  0.0358056   -0.6401598
## 0.0357059  0.0358056    0.0439843   -0.6401598  0.0357992   -0.5963836
## 0.0357059  0.0357992    0.0439843   -0.5963836  0.0357928   -0.5525931
## 0.0357059  0.0357928    0.0439843   -0.5525931  0.0357864   -0.5087884
## 0.0357059  0.0357864    0.0439843   -0.5087884  0.0357800   -0.4649695
## 0.0357059  0.0357800    0.0439843   -0.4649695  0.0357736   -0.4211364
## 0.0357059  0.0357736    0.0439843   -0.4211364  0.0357672   -0.3772891
## 0.0357059  0.0357672    0.0439843   -0.3772891  0.0357608   -0.3334275
## 0.0357059  0.0357608    0.0439843   -0.3334275  0.0357544   -0.2895516
## 0.0357059  0.0357544    0.0439843   -0.2895516  0.0357480   -0.2456615
## 0.0357059  0.0357480    0.0439843   -0.2456615  0.0357417   -0.2017571
## 0.0357059  0.0357417    0.0439843   -0.2017571  0.0357353   -0.1578384
## 0.0357059  0.0357353    0.0439843   -0.1578384  0.0357289   -0.1139055
## 0.0357059  0.0357289    0.0439843   -0.1139055  0.0357225   -0.0699582
## 0.0357059  0.0357225    0.0439843   -0.0699582  0.0357161   -0.0259966
## 0.0357059  0.0357161    0.0439843   -0.0259966  0.0357097    0.0179792
## 0.0357097  0.0357161    0.0179792   -0.0259966  0.0357135   -0.0080209
## 0.0357097  0.0357135    0.0179792   -0.0080209  0.0357109    0.0099573
## 0.0357109  0.0357135    0.0099573   -0.0080209  0.0357120    0.0019358
## 0.0357120  0.0357135    0.0019358   -0.0080209  0.0357132   -0.0060852
## 0.0357120  0.0357132    0.0019358   -0.0060852  0.0357129   -0.0041495
## 0.0357120  0.0357129    0.0019358   -0.0041495  0.0357126   -0.0022138
## =========  =========  ===========  ===========  =========  ===========
Ridder Method
a b func.a func.b d func.d
0.0200000 0.0500000 185.4187715 -72.7792041 0.0343561 9.6653759
0.0343561 0.0500000 9.6653759 -72.7792041 0.0486078 -67.3340615
0.0343561 0.0486078 9.6653759 -67.3340615 0.0472218 -61.6463651
0.0343561 0.0472218 9.6653759 -61.6463651 0.0458414 -55.6935156
0.0343561 0.0458414 9.6653759 -55.6935156 0.0444663 -49.4499989
0.0343561 0.0444663 9.6653759 -49.4499989 0.0430959 -42.8868806
0.0343561 0.0430959 9.6653759 -42.8868806 0.0417296 -35.9711896
0.0343561 0.0417296 9.6653759 -35.9711896 0.0403668 -28.6651600
0.0343561 0.0403668 9.6653759 -28.6651600 0.0390069 -20.9252915
0.0343561 0.0390069 9.6653759 -20.9252915 0.0376491 -12.7011749
0.0343561 0.0376491 9.6653759 -12.7011749 0.0362926 -3.9340090
0.0343561 0.0362926 9.6653759 -3.9340090 0.0349365 5.4452895
0.0349365 0.0362926 5.4452895 -3.9340090 0.0355168 1.3520540
0.0355168 0.0362926 1.3520540 -3.9340090 0.0360970 -2.6210252
0.0355168 0.0360970 1.3520540 -2.6210252 0.0359015 -1.2951049
0.0355168 0.0359015 1.3520540 -1.2951049 0.0357059 0.0439843
0.0357059 0.0359015 0.0439843 -1.2951049 0.0358951 -1.2515407
0.0357059 0.0358951 0.0439843 -1.2515407 0.0358887 -1.2079624
0.0357059 0.0358887 0.0439843 -1.2079624 0.0358823 -1.1643700
0.0357059 0.0358823 0.0439843 -1.1643700 0.0358759 -1.1207636
0.0357059 0.0358759 0.0439843 -1.1207636 0.0358695 -1.0771430
0.0357059 0.0358695 0.0439843 -1.0771430 0.0358631 -1.0335083
0.0357059 0.0358631 0.0439843 -1.0335083 0.0358567 -0.9898595
0.0357059 0.0358567 0.0439843 -0.9898595 0.0358503 -0.9461966
0.0357059 0.0358503 0.0439843 -0.9461966 0.0358439 -0.9025196
0.0357059 0.0358439 0.0439843 -0.9025196 0.0358375 -0.8588283
0.0357059 0.0358375 0.0439843 -0.8588283 0.0358311 -0.8151230
0.0357059 0.0358311 0.0439843 -0.8151230 0.0358247 -0.7714035
0.0357059 0.0358247 0.0439843 -0.7714035 0.0358184 -0.7276698
0.0357059 0.0358184 0.0439843 -0.7276698 0.0358120 -0.6839219
0.0357059 0.0358120 0.0439843 -0.6839219 0.0358056 -0.6401598
0.0357059 0.0358056 0.0439843 -0.6401598 0.0357992 -0.5963836
0.0357059 0.0357992 0.0439843 -0.5963836 0.0357928 -0.5525931
0.0357059 0.0357928 0.0439843 -0.5525931 0.0357864 -0.5087884
0.0357059 0.0357864 0.0439843 -0.5087884 0.0357800 -0.4649695
0.0357059 0.0357800 0.0439843 -0.4649695 0.0357736 -0.4211364
0.0357059 0.0357736 0.0439843 -0.4211364 0.0357672 -0.3772891
0.0357059 0.0357672 0.0439843 -0.3772891 0.0357608 -0.3334275
0.0357059 0.0357608 0.0439843 -0.3334275 0.0357544 -0.2895516
0.0357059 0.0357544 0.0439843 -0.2895516 0.0357480 -0.2456615
0.0357059 0.0357480 0.0439843 -0.2456615 0.0357417 -0.2017571
0.0357059 0.0357417 0.0439843 -0.2017571 0.0357353 -0.1578384
0.0357059 0.0357353 0.0439843 -0.1578384 0.0357289 -0.1139055
0.0357059 0.0357289 0.0439843 -0.1139055 0.0357225 -0.0699582
0.0357059 0.0357225 0.0439843 -0.0699582 0.0357161 -0.0259966
0.0357059 0.0357161 0.0439843 -0.0259966 0.0357097 0.0179792
0.0357097 0.0357161 0.0179792 -0.0259966 0.0357135 -0.0080209
0.0357097 0.0357135 0.0179792 -0.0080209 0.0357109 0.0099573
0.0357109 0.0357135 0.0099573 -0.0080209 0.0357120 0.0019358
0.0357120 0.0357135 0.0019358 -0.0080209 0.0357132 -0.0060852
0.0357120 0.0357132 0.0019358 -0.0060852 0.0357129 -0.0041495
0.0357120 0.0357129 0.0019358 -0.0041495 0.0357126 -0.0022138

Uniroot by R

# uniroot
uniroot(f,c(0.02,0.05),tol = 10^(-6))
## $root
## [1] 0.03571253
## 
## $f.root
## [1] -0.001566892
## 
## $iter
## [1] 6
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 0.0000005

Newton

Newton(f,0.05, tol = 0.000001)
## [1] "Newton's Method "
## [1] "covegence criterion : tolerence <1e-6"
## [1] "number of iterations :  6"
## [1] "approximation root : 0.035712302240628"
## 
## 
## =========  =========  ===========
## Iterative          x         f(x)
## =========  =========  ===========
##         0  0.0500000  -72.7792041
##         1  0.0309488   37.3612136
##         2  0.0351208    4.1317228
##         3  0.0357034    0.0614993
##         4  0.0357123    0.0000140
##         5  0.0357123    0.0000000
## =========  =========  ===========
Newton-Raphosn Method
Iterative x f(x)
0 0.0500000 -72.7792041
1 0.0309488 37.3612136
2 0.0351208 4.1317228
3 0.0357034 0.0614993
4 0.0357123 0.0000140
5 0.0357123 0.0000000

Ridder method by R

# ridder
ridders(f,0.02,0.05)
## $root
## [1] 0.0357123
## 
## $f.root
## [1] -0.00000000000005684342
## 
## $niter
## [1] 10
## 
## $estim.prec
## [1] 0.000000000001726154

Secant.method

# Secant
secant.method(f,0.02,0.05)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  6"
## [1] "approximation root : 0.035712302284155"
## 
## 
## =========  =========  ===========  ===========  ===========  ===========
##         a          b            c       func.a       func.b       func.c
## =========  =========  ===========  ===========  ===========  ===========
## 0.0200000  0.0500000  185.4187715  -72.7792041    0.0415438  -35.0001622
## 0.0200000  0.0500000    0.0439274  185.4187715  -72.7792041  -46.9114689
## 0.0500000  0.0415438    0.0264516  -72.7792041  -35.0001622   84.0107653
## 0.0415438  0.0337096    0.0353333  -35.0001622   14.5228514    2.6330696
## 0.0337096  0.0360070    0.0357614   14.5228514   -2.0123962   -0.3375586
## 0.0360070  0.0357274    0.0357114   -2.0123962   -0.1038302    0.0064722
## 0.0357274  0.0357122    0.0357123   -0.1038302    0.0007789    0.0000055
## =========  =========  ===========  ===========  ===========  ===========
Secant Method
a b c func.a func.b func.c
0.0200000 0.0500000 185.4187715 -72.7792041 0.0415438 -35.0001622
0.0200000 0.0500000 0.0439274 185.4187715 -72.7792041 -46.9114689
0.0500000 0.0415438 0.0264516 -72.7792041 -35.0001622 84.0107653
0.0415438 0.0337096 0.0353333 -35.0001622 14.5228514 2.6330696
0.0337096 0.0360070 0.0357614 14.5228514 -2.0123962 -0.3375586
0.0360070 0.0357274 0.0357114 -2.0123962 -0.1038302 0.0064722
0.0357274 0.0357122 0.0357123 -0.1038302 0.0007789 0.0000055

Bisection

# Bisection
bisection.method(f,0.02,0.05)
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  12"
## [1] "approximation root : 0.0357177734375"
## 
## 
## =========  =========  ===========  ===========  =========  ===========
##         a          b       func.a       func.b          c       func.c
## =========  =========  ===========  ===========  =========  ===========
## 0.0200000  0.0500000  185.4187715  -72.7792041  0.0350000    4.9912068
## 0.0350000  0.0500000    4.9912068  -72.7792041  0.0425000  -39.9187177
## 0.0350000  0.0425000    4.9912068  -39.9187177  0.0387500  -19.4091680
## 0.0350000  0.0387500    4.9912068  -19.4091680  0.0368750   -7.7697759
## 0.0350000  0.0368750    4.9912068   -7.7697759  0.0359375   -1.5404876
## 0.0350000  0.0359375    4.9912068   -1.5404876  0.0354688    1.6860561
## 0.0354688  0.0359375    1.6860561   -1.5404876  0.0357031    0.0631524
## 0.0357031  0.0359375    0.0631524   -1.5404876  0.0358203   -0.7410519
## 0.0357031  0.0358203    0.0631524   -0.7410519  0.0357617   -0.3395487
## 0.0357031  0.0357617    0.0631524   -0.3395487  0.0357324   -0.1383483
## 0.0357031  0.0357324    0.0631524   -0.1383483  0.0357178   -0.0376355
## 0.0357031  0.0357178    0.0631524   -0.0376355  0.0357104    0.0127490
## 0.0357104  0.0357178    0.0127490   -0.0376355  0.0357141   -0.0124456
## =========  =========  ===========  ===========  =========  ===========
Bisection Method
a b func.a func.b c func.c
0.0200000 0.0500000 185.4187715 -72.7792041 0.0350000 4.9912068
0.0350000 0.0500000 4.9912068 -72.7792041 0.0425000 -39.9187177
0.0350000 0.0425000 4.9912068 -39.9187177 0.0387500 -19.4091680
0.0350000 0.0387500 4.9912068 -19.4091680 0.0368750 -7.7697759
0.0350000 0.0368750 4.9912068 -7.7697759 0.0359375 -1.5404876
0.0350000 0.0359375 4.9912068 -1.5404876 0.0354688 1.6860561
0.0354688 0.0359375 1.6860561 -1.5404876 0.0357031 0.0631524
0.0357031 0.0359375 0.0631524 -1.5404876 0.0358203 -0.7410519
0.0357031 0.0358203 0.0631524 -0.7410519 0.0357617 -0.3395487
0.0357031 0.0357617 0.0631524 -0.3395487 0.0357324 -0.1383483
0.0357031 0.0357324 0.0631524 -0.1383483 0.0357178 -0.0376355
0.0357031 0.0357178 0.0631524 -0.0376355 0.0357104 0.0127490
0.0357104 0.0357178 0.0127490 -0.0376355 0.0357141 -0.0124456

false position

# false position 
false.position.method(f,0.02,0.05)
## [1] "covegence criterion : |Cn-Cn-1|<1e-5"
## [1] "number of iterations :  9"
## [1] "approximation root : 0.0357175557046726"
## 
## 
## ====  =========  =========  ========  ===========  ===========
##    a          b          c    func.a       func.b       func.c
## ====  =========  =========  ========  ===========  ===========
## 0.02  0.0415438  0.0415438  185.4188  -35.0001622  -35.0001622
## 0.02  0.0381229  0.0381229  185.4188  -15.6307120  -15.6307120
## 0.02  0.0367139  0.0367139  185.4188   -6.7198107   -6.7198107
## 0.02  0.0361293  0.0361293  185.4188   -2.8390660   -2.8390660
## 0.02  0.0358861  0.0358861  185.4188   -1.1904584   -1.1904584
## 0.02  0.0357848  0.0357848  185.4188   -0.4975792   -0.4975792
## 0.02  0.0357425  0.0357425  185.4188   -0.2076951   -0.2076951
## 0.02  0.0357249  0.0357249  185.4188   -0.0866455   -0.0866455
## 0.02  0.0357176  0.0357176  185.4188   -0.0361380   -0.0361380
## ====  =========  =========  ========  ===========  ===========
False Position Method
a b c func.a func.b func.c
0.02 0.0415438 0.0415438 185.4188 -35.0001622 -35.0001622
0.02 0.0381229 0.0381229 185.4188 -15.6307120 -15.6307120
0.02 0.0367139 0.0367139 185.4188 -6.7198107 -6.7198107
0.02 0.0361293 0.0361293 185.4188 -2.8390660 -2.8390660
0.02 0.0358861 0.0358861 185.4188 -1.1904584 -1.1904584
0.02 0.0357848 0.0357848 185.4188 -0.4975792 -0.4975792
0.02 0.0357425 0.0357425 185.4188 -0.2076951 -0.2076951
0.02 0.0357249 0.0357249 185.4188 -0.0866455 -0.0866455
0.02 0.0357176 0.0357176 185.4188 -0.0361380 -0.0361380

判斷是否為極大值

# s.o.c
grad(f,0.0357123) < 0
## [1] TRUE
Do you ever shine?