Soal

knitr::include_graphics("Tugas Praktikum 10.JPG")

No 1

Poin a

knitr::include_graphics("Nomor 1_page-1.JPG")

knitr::include_graphics("Nomor 1_page-2.JPG")

Poin b

nilai exact dari E(X) adalah 0.710227

a <- 5
b <- 3
Ex <- (b*gamma(1+1/a)*gamma(b))/gamma(1+1/a+b)
Ex

## [1] 0.7102273

Dengan nilai toleransi = 0.0001

Metode Trapezoidal

fungsi trapezoidal

trapezoid <- function(ftn, a, b, n = 100) {
     h <- (b-a)/n
     x.vec <- seq(a, b, by = h)
     f.vec <- sapply(x.vec, ftn)     # ftn(x.vec)
     Trap <- h*(f.vec[1]/2 + sum(f.vec[2:n]) + f.vec[n+1]/2)
     return(Trap)
}

Mendefinisikan fungsi

E_mt <- function(x){
  (15*(x^5))*((1-(x^5))^2)
}

Menghitung nilai ekspektasi

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_trap <- trapezoid(E_mt,0,1,n = n)
  
  err <- abs(res_trap-exact_value)
  
  cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.6312869, error=0.07894012
## n=5, result=0.6738123, error=0.03641467
## n=6, result=0.6914928, error=0.01873424
## n=7, result=0.6997126, error=0.01051443
## n=8, result=0.7039057, error=0.006321303
## n=9, result=0.7062116, error=0.004015356
## n=10, result=0.7075597, error=0.002667293
## n=11, result=0.7083885, error=0.001838522
## n=12, result=0.7089199, error=0.001307139
## n=13, result=0.7092729, error=0.0009541148
## n=14, result=0.7095147, error=0.000712349
## n=15, result=0.7096846, error=0.0005423739
## n=16, result=0.7098069, error=0.0004201042
## n=17, result=0.7098966, error=0.0003303616
## n=18, result=0.7099637, error=0.0002633072
## n=19, result=0.7100146, error=0.0002124014
## n=20, result=0.7100538, error=0.0001731994
## n=21, result=0.7100844, error=0.0001426188
## n=22, result=0.7101085, error=0.0001184833
## n=23, result=0.7101278, error=9.923088e-05

Dengan metode Trapezoidal diperoleh nilai E(X) sebesar 0.7101278 pada iterasi ke 23 dengan nlai error = 9.92308e-05

Metode Simpson

Fungsi Simpson

simpson_n <- function(ftn, a, b, n = 100) {
    n <- max(c(2*(n %/% 2), 4))
    h <- (b-a)/n
    x.vec1 <- seq(a+h, b-h, by = 2*h)       # ganjil
    x.vec2 <- seq(a+2*h, b-2*h, by = 2*h)   # genap
    f.vec1 <- sapply(x.vec1, ftn)   # ganjil
    f.vec2 <- sapply(x.vec2, ftn)   # genap
    S <- h/3*(ftn(a) + ftn(b) + 4*sum(f.vec1) + 2*sum(f.vec2))
    return(S)
}

Mendefinisikan fungsi

E_ms <- function(x){
  (15*(x^5))*((1-(x^5))^2)
}

Menghitung nilai ekspektasi

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_simp <- simpson_n(E_ms ,0,1,n = n)
  
  err <- abs(res_simp-exact_value)
  
  cat("n=",n,", result=",res_simp,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.7683974, error=0.05817036
## n=5, result=0.7683974, error=0.05817036
## n=6, result=0.7497082, error=0.03948124
## n=7, result=0.7497082, error=0.03948124
## n=8, result=0.728112, error=0.01788497
## n=9, result=0.728112, error=0.01788497
## n=10, result=0.7188088, error=0.008581832
## n=11, result=0.7188088, error=0.008581832
## n=12, result=0.7147289, error=0.004501893
## n=13, result=0.7147289, error=0.004501893
## n=14, result=0.712782, error=0.002555012
## n=15, result=0.712782, error=0.002555012
## n=16, result=0.711774, error=0.001546962
## n=17, result=0.711774, error=0.001546962
## n=18, result=0.7112144, error=0.0009873756
## n=19, result=0.7112144, error=0.0009873756
## n=20, result=0.7108852, error=0.0006581651
## n=21, result=0.7108852, error=0.0006581651
## n=22, result=0.7106819, error=0.0004548628
## n=23, result=0.7106819, error=0.0004548628
## n=24, result=0.7105511, error=0.0003240819
## n=25, result=0.7105511, error=0.0003240819
## n=26, result=0.710464, error=0.0002369803
## n=27, result=0.710464, error=0.0002369803
## n=28, result=0.7104042, error=0.0001772138
## n=29, result=0.7104042, error=0.0001772138
## n=30, result=0.7103621, error=0.0001351299
## n=31, result=0.7103621, error=0.0001351299
## n=32, result=0.7103318, error=0.0001048198
## n=33, result=0.7103318, error=0.0001048198
## n=34, result=0.7103096, error=8.255045e-05

Denga metode Simpson diperoleh nilai E(X) sebesar 0.7103096 pada iterasi ke 34 dan nilai error sebesar = 8.255045e-05

Metode Four - Point Gauss Quadrature

library(pracma)

## Warning: package 'pracma' was built under R version 4.2.3

Mencari nilai ekspektasi

E_mfpgq <- function(x){
  (15*(x^5))*((1-(x^5))^2)
}

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = 0,1)

 Ci <- gL$w # koefisien
 xi <- gL$x # gauss point
  
  res_gl <- sum(Ci * E_mfpgq(xi))

  err <- abs(res_gl-exact_value)
  
  cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.6792999, error=0.03092708
## n=5, result=0.707375, error=0.002851979
## n=6, result=0.7101422, error=8.477726e-05

Dengan metode Four - Point Gauss Quadrature diperoleh nilai E(X) sebesar 0.7101422 pada iterasi ke 6 dan nilai error sebesar = 8.477726e-05

Membandingkan ketiga metode

Et <- 0.7101278
Es <- 0.7103096
Efpgq <- 0.7101422
errt <- abs(Et-Ex)
errs <- abs(Es-Ex)
errfpgq <- abs(Efpgq-Ex)

Metode <- c("Trapezoidal","Simpson","Four-Point Gauss Quadrature")
Iterasi <- c(23,34,6)
Ekpektasi <- c(Et,Es,Efpgq)
Error <- c(errt,errs,errfpgq)

Perbandingan.metode <- data.frame("Metode"=Metode, "Iterasi"=Iterasi, "E(X)"=Ekpektasi, "Error"=Error)
Perbandingan.metode

##                        Metode Iterasi      E.X.        Error
## 1                 Trapezoidal      23 0.7101278 9.947273e-05
## 2                     Simpson      34 0.7103096 8.232727e-05
## 3 Four-Point Gauss Quadrature       6 0.7101422 8.507273e-05

Berdasarkan ketiga metode di atas, nilai E(X) yang paling mendekati nilai exact nya adalah metode Simpson.

No 2

Dengan nilai Toleransi = 0.00001 n = 4 nilai exact = 0.99966

f <- function(x) {1/2*exp(-1/2*x)}
nilai.exact <- integrate(f,lower=0,upper=Inf)
nilai.exact

## 1 with absolute error < 3.4e-05

Metode Trapezoidal

Mendefinisikan fungsi

f <- function(x) {
  2*exp(-2*x)
}

Menghitung cdf

exact_value=0.99966
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_trap <- trapezoid(f,0,4,n = n)
  
  err <- abs(res_trap-exact_value)
  
  cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=1.312595, error=0.3129348
## n=5, result=1.204348, error=0.2046884
## n=6, result=1.143553, error=0.1438927
## n=7, result=1.106174, error=0.1065143
## n=8, result=1.081614, error=0.08195374
## n=9, result=1.064635, error=0.06497528
## n=10, result=1.05242, error=0.05275981
## n=11, result=1.043343, error=0.04368329
## n=12, result=1.036418, error=0.03675776
## n=13, result=1.031015, error=0.0313548
## n=14, result=1.026719, error=0.0270594
## n=15, result=1.023249, error=0.02358871
## n=16, result=1.020405, error=0.02074462
## n=17, result=1.018045, error=0.01838505
## n=18, result=1.016066, error=0.016406
## n=19, result=1.01439, error=0.0147299
## n=20, result=1.012958, error=0.01329799
## n=21, result=1.011725, error=0.01206507
## n=22, result=1.010656, error=0.01099592
## n=23, result=1.009723, error=0.01006281
## n=24, result=1.008904, error=0.009243595
## n=25, result=1.00818, error=0.008520485
## n=26, result=1.007539, error=0.00787902
## n=27, result=1.006967, error=0.007307362
## n=28, result=1.006456, error=0.006795742
## n=29, result=1.005996, error=0.006336041
## n=30, result=1.005581, error=0.005921466
## n=31, result=1.005206, error=0.005546302
## n=32, result=1.004866, error=0.005205708
## n=33, result=1.004556, error=0.004895565
## n=34, result=1.004272, error=0.00461235
## n=35, result=1.004013, error=0.004353033
## n=36, result=1.003775, error=0.004115001
## n=37, result=1.003556, error=0.003895987
## n=38, result=1.003354, error=0.003694018
## n=39, result=1.003167, error=0.00350737
## n=40, result=1.002995, error=0.003334533
## n=41, result=1.002834, error=0.003174177
## n=42, result=1.002685, error=0.003025129
## n=43, result=1.002546, error=0.00288635
## n=44, result=1.002417, error=0.002756918
## n=45, result=1.002296, error=0.002636013
## n=46, result=1.002183, error=0.002522902
## n=47, result=1.002077, error=0.002416928
## n=48, result=1.001978, error=0.002317505
## n=49, result=1.001884, error=0.002224103
## n=50, result=1.001796, error=0.002136246
## n=51, result=1.001714, error=0.002053503
## n=52, result=1.001635, error=0.001975485
## n=53, result=1.001562, error=0.001901839
## n=54, result=1.001492, error=0.001832245
## n=55, result=1.001426, error=0.00176641
## n=56, result=1.001364, error=0.001704069
## n=57, result=1.001305, error=0.001644979
## n=58, result=1.001249, error=0.001588917
## n=59, result=1.001196, error=0.001535681
## n=60, result=1.001145, error=0.001485083
## n=61, result=1.001097, error=0.001436952
## n=62, result=1.001051, error=0.001391131
## n=63, result=1.001007, error=0.001347473
## n=64, result=1.000966, error=0.001305845
## n=65, result=1.000926, error=0.001266123
## n=66, result=1.000888, error=0.001228192
## n=67, result=1.000852, error=0.001191946
## n=68, result=1.000817, error=0.001157287
## n=69, result=1.000784, error=0.001124124
## n=70, result=1.000752, error=0.001092371
## n=71, result=1.000722, error=0.00106195
## n=72, result=1.000693, error=0.001032787
## n=73, result=1.000665, error=0.001004815
## n=74, result=1.000638, error=0.000977968
## n=75, result=1.000612, error=0.0009521878
## n=76, result=1.000587, error=0.0009274182
## n=77, result=1.000564, error=0.0009036072
## n=78, result=1.000541, error=0.000880706
## n=79, result=1.000519, error=0.0008586686
## n=80, result=1.000497, error=0.0008374523
## n=81, result=1.000477, error=0.0008170168
## n=82, result=1.000457, error=0.0007973242
## n=83, result=1.000438, error=0.0007783389
## n=84, result=1.00042, error=0.0007600275
## n=85, result=1.000402, error=0.0007423584
## n=86, result=1.000385, error=0.000725302
## n=87, result=1.000369, error=0.0007088303
## n=88, result=1.000353, error=0.0006929168
## n=89, result=1.000338, error=0.0006775365
## n=90, result=1.000323, error=0.000662666
## n=91, result=1.000308, error=0.000648283
## n=92, result=1.000294, error=0.0006343663
## n=93, result=1.000281, error=0.0006208961
## n=94, result=1.000268, error=0.0006078534
## n=95, result=1.000255, error=0.0005952204
## n=96, result=1.000243, error=0.00058298
## n=97, result=1.000231, error=0.0005711162
## n=98, result=1.00022, error=0.0005596136
## n=99, result=1.000208, error=0.0005484578
## n=100, result=1.000198, error=0.0005376349
## n=101, result=1.000187, error=0.0005271319
## n=102, result=1.000177, error=0.0005169362
## n=103, result=1.000167, error=0.000507036
## n=104, result=1.000157, error=0.00049742
## n=105, result=1.000148, error=0.0004880774
## n=106, result=1.000139, error=0.0004789979
## n=107, result=1.00013, error=0.0004701717
## n=108, result=1.000122, error=0.0004615896
## n=109, result=1.000113, error=0.0004532425
## n=110, result=1.000105, error=0.000445122
## n=111, result=1.000097, error=0.00043722
## n=112, result=1.00009, error=0.0004295287
## n=113, result=1.000082, error=0.0004220406
## n=114, result=1.000075, error=0.0004147487
## n=115, result=1.000068, error=0.0004076462
## n=116, result=1.000061, error=0.0004007266
## n=117, result=1.000054, error=0.0003939836
## n=118, result=1.000047, error=0.0003874112
## n=119, result=1.000041, error=0.0003810039
## n=120, result=1.000035, error=0.0003747561
## n=121, result=1.000029, error=0.0003686625
## n=122, result=1.000023, error=0.0003627181
## n=123, result=1.000017, error=0.0003569181
## n=124, result=1.000011, error=0.0003512579
## n=125, result=1.000006, error=0.0003457329
## n=126, result=1, error=0.000340339
## n=127, result=0.9999951, error=0.0003350719
## n=128, result=0.9999899, error=0.0003299278
## n=129, result=0.9999849, error=0.0003249029
## n=130, result=0.99998, error=0.0003199935
## n=131, result=0.9999752, error=0.000315196
## n=132, result=0.9999705, error=0.0003105072
## n=133, result=0.9999659, error=0.0003059237
## n=134, result=0.9999614, error=0.0003014424
## n=135, result=0.9999571, error=0.0002970604
## n=136, result=0.9999528, error=0.0002927747
## n=137, result=0.9999486, error=0.0002885824
## n=138, result=0.9999445, error=0.000284481
## n=139, result=0.9999405, error=0.0002804677
## n=140, result=0.9999365, error=0.0002765401
## n=141, result=0.9999327, error=0.0002726958
## n=142, result=0.9999289, error=0.0002689324
## n=143, result=0.9999252, error=0.0002652477
## n=144, result=0.9999216, error=0.0002616395
## n=145, result=0.9999181, error=0.0002581057
## n=146, result=0.9999146, error=0.0002546442
## n=147, result=0.9999113, error=0.0002512531
## n=148, result=0.9999079, error=0.0002479306
## n=149, result=0.9999047, error=0.0002446747
## n=150, result=0.9999015, error=0.0002414837
## n=151, result=0.9998984, error=0.0002383558
## n=152, result=0.9998953, error=0.0002352895
## n=153, result=0.9998923, error=0.0002322832
## n=154, result=0.9998893, error=0.0002293352
## n=155, result=0.9998864, error=0.000226444
## n=156, result=0.9998836, error=0.0002236083
## n=157, result=0.9998808, error=0.0002208266
## n=158, result=0.9998781, error=0.0002180976
## n=159, result=0.9998754, error=0.0002154198
## n=160, result=0.9998728, error=0.0002127921
## n=161, result=0.9998702, error=0.0002102133
## n=162, result=0.9998677, error=0.000207682
## n=163, result=0.9998652, error=0.0002051972
## n=164, result=0.9998628, error=0.0002027577
## n=165, result=0.9998604, error=0.0002003624
## n=166, result=0.999858, error=0.0001980102
## n=167, result=0.9998557, error=0.0001957002
## n=168, result=0.9998534, error=0.0001934313
## n=169, result=0.9998512, error=0.0001912026
## n=170, result=0.999849, error=0.0001890131
## n=171, result=0.9998469, error=0.0001868618
## n=172, result=0.9998447, error=0.000184748
## n=173, result=0.9998427, error=0.0001826708
## n=174, result=0.9998406, error=0.0001806292
## n=175, result=0.9998386, error=0.0001786225
## n=176, result=0.9998366, error=0.00017665
## n=177, result=0.9998347, error=0.0001747108
## n=178, result=0.9998328, error=0.0001728042
## n=179, result=0.9998309, error=0.0001709294
## n=180, result=0.9998291, error=0.0001690858
## n=181, result=0.9998273, error=0.0001672727
## n=182, result=0.9998255, error=0.0001654893
## n=183, result=0.9998237, error=0.0001637352
## n=184, result=0.999822, error=0.0001620095
## n=185, result=0.9998203, error=0.0001603117
## n=186, result=0.9998186, error=0.0001586413
## n=187, result=0.999817, error=0.0001569976
## n=188, result=0.9998154, error=0.00015538
## n=189, result=0.9998138, error=0.0001537881
## n=190, result=0.9998122, error=0.0001522212
## n=191, result=0.9998107, error=0.0001506789
## n=192, result=0.9998092, error=0.0001491606
## n=193, result=0.9998077, error=0.0001476658
## n=194, result=0.9998062, error=0.0001461941
## n=195, result=0.9998047, error=0.000144745
## n=196, result=0.9998033, error=0.000143318
## n=197, result=0.9998019, error=0.0001419127
## n=198, result=0.9998005, error=0.0001405286
## n=199, result=0.9997992, error=0.0001391653
## n=200, result=0.9997978, error=0.0001378224
## n=201, result=0.9997965, error=0.0001364995
## n=202, result=0.9997952, error=0.0001351962
## n=203, result=0.9997939, error=0.0001339122
## n=204, result=0.9997926, error=0.0001326469
## n=205, result=0.9997914, error=0.0001314002
## n=206, result=0.9997902, error=0.0001301715
## n=207, result=0.999789, error=0.0001289606
## n=208, result=0.9997878, error=0.0001277671
## n=209, result=0.9997866, error=0.0001265908
## n=210, result=0.9997854, error=0.0001254311
## n=211, result=0.9997843, error=0.000124288
## n=212, result=0.9997832, error=0.0001231609
## n=213, result=0.999782, error=0.0001220497
## n=214, result=0.999781, error=0.0001209541
## n=215, result=0.9997799, error=0.0001198737
## n=216, result=0.9997788, error=0.0001188083
## n=217, result=0.9997778, error=0.0001177575
## n=218, result=0.9997767, error=0.0001167212
## n=219, result=0.9997757, error=0.0001156991
## n=220, result=0.9997747, error=0.0001146908
## n=221, result=0.9997737, error=0.0001136962
## n=222, result=0.9997727, error=0.0001127151
## n=223, result=0.9997717, error=0.000111747
## n=224, result=0.9997708, error=0.000110792
## n=225, result=0.9997698, error=0.0001098496
## n=226, result=0.9997689, error=0.0001089197
## n=227, result=0.999768, error=0.0001080021
## n=228, result=0.9997671, error=0.0001070965
## n=229, result=0.9997662, error=0.0001062028
## n=230, result=0.9997653, error=0.0001053207
## n=231, result=0.9997644, error=0.00010445
## n=232, result=0.9997636, error=0.0001035906
## n=233, result=0.9997627, error=0.0001027422
## n=234, result=0.9997619, error=0.0001019046
## n=235, result=0.9997611, error=0.0001010777
## n=236, result=0.9997603, error=0.0001002613
## n=237, result=0.9997595, error=9.945526e-05

Pada metode Trapezoidal diperoleh nilai CDF sebesar 0.9997595 pada iterasi ke 237 dan nilai error sebesar = 9,45526-05

Metode Simpson

Fungsi metode Simpson

simpson_n <- function(ftn, a, b, n = 100) {
    n <- max(c(2*(n %/% 2), 4))
    h <- (b-a)/n
    x.vec1 <- seq(a+h, b-h, by = 2*h)       
    x.vec2 <- seq(a+2*h, b-2*h, by = 2*h)   
    f.vec1 <- sapply(x.vec1, ftn)  
    f.vec2 <- sapply(x.vec2, ftn)  
    S <- h/3*(ftn(a) + ftn(b) + 4*sum(f.vec1) + 2*sum(f.vec2))
    return(S)
}

Mendefinisikan fungsi

f <- function(x) {
  2*exp(-2*x)
}

Menhitung nilai CDF

exact_value=0.99966
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_simp <- simpson_n(f,0,4,n = n)
  
  err <- abs(res_simp-exact_value)
  
  cat("n=",n,", result=",res_simp,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=1.058815, error=0.05915525
## n=5, result=1.058815, error=0.05915525
## n=6, result=1.014089, error=0.01442885
## n=7, result=1.014089, error=0.01442885
## n=8, result=1.00462, error=0.004960055
## n=9, result=1.00462, error=0.004960055
## n=10, result=1.001777, error=0.002116944
## n=11, result=1.001777, error=0.002116944
## n=12, result=1.000706, error=0.00104611
## n=13, result=1.000706, error=0.00104611
## n=14, result=1.000234, error=0.0005744298
## n=15, result=1.000234, error=0.0005744298
## n=16, result=1.000002, error=0.000341577
## n=17, result=1.000002, error=0.000341577
## n=18, result=0.9998762, error=0.0002162416
## n=19, result=0.9998762, error=0.0002162416
## n=20, result=0.999804, error=0.0001440486
## n=21, result=0.999804, error=0.0001440486
## n=22, result=0.9997601, error=0.0001001368
## n=23, result=0.9997601, error=0.0001001368
## n=24, result=0.9997322, error=7.2205e-05

Dengan metode Simpson diperoleh nilai CDF sebesar 0.9997322 pada iterasi ke 24 dan nilai error sebesar = 7.2205-05

Metode Four - Point Gauss Quadrature

exact_value=0.99966
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = 0,4)

 Ci <- gL$w # koefisien
 xi <- gL$x # gauss point
  
  res_gl <- sum(Ci * f(xi))

  err <- abs(res_gl-exact_value)
  
  cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.9976267, error=0.002033306
## n=5, result=0.9995785, error=8.14847e-05

Pada Metode Four - Point Gauss Quadrature diperoleh nilai CDF sebesar 0.9995785 pada iterasi ke 5 dan nilai error sebesar = 8.14847e-05

Metode Integral Monte Carlo

Fungsi monte carlo

mc_integral <- function(ftn, a, b,m=1000){
  #Membangkitkan x berdistribusi U(a,b)
  x <- runif(m,a,b)
  # Menghitung rata-rata dari output fungsi
  Gx <- ftn(x)
  Gx_m <- mean(Gx)
  theta.hat <- (b-a)*Gx_m
  return(theta.hat)
}

Mendefinisikan fungsi

f <- function(x) {
  2*exp(-2*x)
}

Menghitung cdf

set.seed(100)
mc_integral(f, a=0, b=4, m=1000)

## [1] 0.9213784

Pada Metode Integral Monte Carlo diperoleh nilai CDF sebesar 0.9213784

Membandingkan ketiga metode

cdft <- 0.9997595
cdfs <- 0.9997322
cdffpgq <- 0.9995785
cdfmc <- 0.9213784
nilai.exact <- 0.99966

errt <- abs(cdft-nilai.exact)
errs <- abs(cdfs-nilai.exact)
errfpgq <- abs(cdffpgq-nilai.exact)
errmc <- abs(cdfmc-nilai.exact)

Metode <- c("Trapezoidal","Simpson","Four-Point Gauss Quadrature","Integral Monte Carlo")
Iterasi <- c(237,24,5," ")
cdf <- c(cdft,cdfs,cdffpgq,cdfmc)
Error <- c(errt,errs,errfpgq,errmc)

Perbandingan.metode <- data.frame("Metode"=Metode, "Iterasi"=Iterasi, "CDF"=cdf, "Error"=Error)
Perbandingan.metode

##                        Metode Iterasi       CDF     Error
## 1                 Trapezoidal     237 0.9997595 0.0000995
## 2                     Simpson      24 0.9997322 0.0000722
## 3 Four-Point Gauss Quadrature       5 0.9995785 0.0000815
## 4        Integral Monte Carlo         0.9213784 0.0782816

Berdasarkan perbandingan keempat metode di atas, dapat dilihat bahwa nilai CDF dari distribusi exp(2) yang paling mendekati nilai exact nya atau error terkecil yaitu dengan metode Simpson yaitu 0.9997322 pada iterasi ke 24

Tugas 10 Pemograman Statistika

2022-11-14

Soal

No 1

Poin a

Poin b

Metode Trapezoidal

Metode Simpson

Metode Four - Point Gauss Quadrature

No 2

Metode Trapezoidal

Metode Simpson

Metode Four - Point Gauss Quadrature

Metode Integral Monte Carlo

Membandingkan ketiga metode