Bagian a

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

trapezoid(f,0,1,n = 4)

## [1] 0.6312869

trapzfun(f,0,1,maxit =  4)

## $value
## [1] 0.7098069
## 
## $iter
## [1] 4
## 
## $rel.err
## [1] 0.005901199

simpson_n(f,0,1,n = 4)

## [1] 0.7683974

gL <- gaussLegendre(n = 4,a = 0,1)
Ci <- gL$w # koefisien
Ci

## [1] 0.1739274 0.3260726 0.3260726 0.1739274

xi <- gL$x # gauss point
xi

## [1] 0.06943184 0.33000948 0.66999052 0.93056816

I <- sum(Ci * f(xi))
I

## [1] 0.6792999

Bagian b

Jika diketahui nilai exact E(X) \(E(X)= \frac {b \Gamma (1+1/2)\Gamma(b)}{\Gamma(1+(1/a) +b)}\), akan di cari nilai E(X) dengan toleransi 0,0001.

#nilai exact E(X)
a = 5
b = 3
E<-(b*gamma (1+(1/a))*gamma(b))/(gamma(1+(1/a)+b))
E

## [1] 0.7102273

exact_value=0.7102273
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_trap <- trapezoid(f,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=0.07894042
## n=5, result=0.6738123, error=0.03641497
## n=6, result=0.6914928, error=0.01873454
## n=7, result=0.6997126, error=0.01051473
## n=8, result=0.7039057, error=0.006321603
## n=9, result=0.7062116, error=0.004015656
## n=10, result=0.7075597, error=0.002667593
## n=11, result=0.7083885, error=0.001838822
## n=12, result=0.7089199, error=0.001307439
## n=13, result=0.7092729, error=0.0009544148
## n=14, result=0.7095147, error=0.000712649
## n=15, result=0.7096846, error=0.0005426739
## n=16, result=0.7098069, error=0.0004204042
## n=17, result=0.7098966, error=0.0003306616
## n=18, result=0.7099637, error=0.0002636072
## n=19, result=0.7100146, error=0.0002127014
## n=20, result=0.7100538, error=0.0001734994
## n=21, result=0.7100844, error=0.0001429188
## n=22, result=0.7101085, error=0.0001187833
## n=23, result=0.7101278, error=9.953088e-05

exact_value=0.7102273
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_simp <- simpson_n(f,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=0.05817006
## n=5, result=0.7683974, error=0.05817006
## n=6, result=0.7497082, error=0.03948094
## n=7, result=0.7497082, error=0.03948094
## n=8, result=0.728112, error=0.01788467
## n=9, result=0.728112, error=0.01788467
## n=10, result=0.7188088, error=0.008581532
## n=11, result=0.7188088, error=0.008581532
## n=12, result=0.7147289, error=0.004501593
## n=13, result=0.7147289, error=0.004501593
## n=14, result=0.712782, error=0.002554712
## n=15, result=0.712782, error=0.002554712
## n=16, result=0.711774, error=0.001546662
## n=17, result=0.711774, error=0.001546662
## n=18, result=0.7112144, error=0.0009870756
## n=19, result=0.7112144, error=0.0009870756
## n=20, result=0.7108852, error=0.0006578651
## n=21, result=0.7108852, error=0.0006578651
## n=22, result=0.7106819, error=0.0004545628
## n=23, result=0.7106819, error=0.0004545628
## n=24, result=0.7105511, error=0.0003237819
## n=25, result=0.7105511, error=0.0003237819
## n=26, result=0.710464, error=0.0002366803
## n=27, result=0.710464, error=0.0002366803
## n=28, result=0.7104042, error=0.0001769138
## n=29, result=0.7104042, error=0.0001769138
## n=30, result=0.7103621, error=0.0001348299
## n=31, result=0.7103621, error=0.0001348299
## n=32, result=0.7103318, error=0.0001045198
## n=33, result=0.7103318, error=0.0001045198
## n=34, result=0.7103096, error=8.225045e-05

exact_value=0.7102273
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 * 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=0.6792999, error=0.03092738
## n=5, result=0.707375, error=0.002852279
## n=6, result=0.7101422, error=8.507726e-05

Nilai exact CDF

a = 0
b= 4
Ft<- function(x){
  2*exp(-2*x)
}
F <- integrate(Ft,a,b)
F

## 0.9996645 with absolute error < 1.1e-14

diperoleh nilai Exact CDF yaitu 0,9996645.

Dilakukan perhitungan dengan metode-metode berikut:

Metode trapezoida

trapezoid(f1,0,4,n = 9)

## [1] 1.064635

trapzfun(f1,0,4,maxit =  9)

## $value
## [1] 0.9996849
## 
## $iter
## [1] 9
## 
## $rel.err
## [1] 6.101344e-05

exact_value=0.9996645
tol <-  0.00001
err <- 1
n = 9

while(err>tol){
  res_trap <- trapezoid(f1,0,4,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=9, result=1.064635, error=0.06497078
## n=10, result=1.05242, error=0.05275531
## n=11, result=1.043343, error=0.04367879
## n=12, result=1.036418, error=0.03675326
## n=13, result=1.031015, error=0.0313503
## n=14, result=1.026719, error=0.0270549
## n=15, result=1.023249, error=0.02358421
## n=16, result=1.020405, error=0.02074012
## n=17, result=1.018045, error=0.01838055
## n=18, result=1.016066, error=0.0164015
## n=19, result=1.01439, error=0.0147254
## n=20, result=1.012958, error=0.01329349
## n=21, result=1.011725, error=0.01206057
## n=22, result=1.010656, error=0.01099142
## n=23, result=1.009723, error=0.01005831
## n=24, result=1.008904, error=0.009239095
## n=25, result=1.00818, error=0.008515985
## n=26, result=1.007539, error=0.00787452
## n=27, result=1.006967, error=0.007302862
## n=28, result=1.006456, error=0.006791242
## n=29, result=1.005996, error=0.006331541
## n=30, result=1.005581, error=0.005916966
## n=31, result=1.005206, error=0.005541802
## n=32, result=1.004866, error=0.005201208
## n=33, result=1.004556, error=0.004891065
## n=34, result=1.004272, error=0.00460785
## n=35, result=1.004013, error=0.004348533
## n=36, result=1.003775, error=0.004110501
## n=37, result=1.003556, error=0.003891487
## n=38, result=1.003354, error=0.003689518
## n=39, result=1.003167, error=0.00350287
## n=40, result=1.002995, error=0.003330033
## n=41, result=1.002834, error=0.003169677
## n=42, result=1.002685, error=0.003020629
## n=43, result=1.002546, error=0.00288185
## n=44, result=1.002417, error=0.002752418
## n=45, result=1.002296, error=0.002631513
## n=46, result=1.002183, error=0.002518402
## n=47, result=1.002077, error=0.002412428
## n=48, result=1.001978, error=0.002313005
## n=49, result=1.001884, error=0.002219603
## n=50, result=1.001796, error=0.002131746
## n=51, result=1.001714, error=0.002049003
## n=52, result=1.001635, error=0.001970985
## n=53, result=1.001562, error=0.001897339
## n=54, result=1.001492, error=0.001827745
## n=55, result=1.001426, error=0.00176191
## n=56, result=1.001364, error=0.001699569
## n=57, result=1.001305, error=0.001640479
## n=58, result=1.001249, error=0.001584417
## n=59, result=1.001196, error=0.001531181
## n=60, result=1.001145, error=0.001480583
## n=61, result=1.001097, error=0.001432452
## n=62, result=1.001051, error=0.001386631
## n=63, result=1.001007, error=0.001342973
## n=64, result=1.000966, error=0.001301345
## n=65, result=1.000926, error=0.001261623
## n=66, result=1.000888, error=0.001223692
## n=67, result=1.000852, error=0.001187446
## n=68, result=1.000817, error=0.001152787
## n=69, result=1.000784, error=0.001119624
## n=70, result=1.000752, error=0.001087871
## n=71, result=1.000722, error=0.00105745
## n=72, result=1.000693, error=0.001028287
## n=73, result=1.000665, error=0.001000315
## n=74, result=1.000638, error=0.000973468
## n=75, result=1.000612, error=0.0009476878
## n=76, result=1.000587, error=0.0009229182
## n=77, result=1.000564, error=0.0008991072
## n=78, result=1.000541, error=0.000876206
## n=79, result=1.000519, error=0.0008541686
## n=80, result=1.000497, error=0.0008329523
## n=81, result=1.000477, error=0.0008125168
## n=82, result=1.000457, error=0.0007928242
## n=83, result=1.000438, error=0.0007738389
## n=84, result=1.00042, error=0.0007555275
## n=85, result=1.000402, error=0.0007378584
## n=86, result=1.000385, error=0.000720802
## n=87, result=1.000369, error=0.0007043303
## n=88, result=1.000353, error=0.0006884168
## n=89, result=1.000338, error=0.0006730365
## n=90, result=1.000323, error=0.000658166
## n=91, result=1.000308, error=0.000643783
## n=92, result=1.000294, error=0.0006298663
## n=93, result=1.000281, error=0.0006163961
## n=94, result=1.000268, error=0.0006033534
## n=95, result=1.000255, error=0.0005907204
## n=96, result=1.000243, error=0.00057848
## n=97, result=1.000231, error=0.0005666162
## n=98, result=1.00022, error=0.0005551136
## n=99, result=1.000208, error=0.0005439578
## n=100, result=1.000198, error=0.0005331349
## n=101, result=1.000187, error=0.0005226319
## n=102, result=1.000177, error=0.0005124362
## n=103, result=1.000167, error=0.000502536
## n=104, result=1.000157, error=0.00049292
## n=105, result=1.000148, error=0.0004835774
## n=106, result=1.000139, error=0.0004744979
## n=107, result=1.00013, error=0.0004656717
## n=108, result=1.000122, error=0.0004570896
## n=109, result=1.000113, error=0.0004487425
## n=110, result=1.000105, error=0.000440622
## n=111, result=1.000097, error=0.00043272
## n=112, result=1.00009, error=0.0004250287
## n=113, result=1.000082, error=0.0004175406
## n=114, result=1.000075, error=0.0004102487
## n=115, result=1.000068, error=0.0004031462
## n=116, result=1.000061, error=0.0003962266
## n=117, result=1.000054, error=0.0003894836
## n=118, result=1.000047, error=0.0003829112
## n=119, result=1.000041, error=0.0003765039
## n=120, result=1.000035, error=0.0003702561
## n=121, result=1.000029, error=0.0003641625
## n=122, result=1.000023, error=0.0003582181
## n=123, result=1.000017, error=0.0003524181
## n=124, result=1.000011, error=0.0003467579
## n=125, result=1.000006, error=0.0003412329
## n=126, result=1, error=0.000335839
## n=127, result=0.9999951, error=0.0003305719
## n=128, result=0.9999899, error=0.0003254278
## n=129, result=0.9999849, error=0.0003204029
## n=130, result=0.99998, error=0.0003154935
## n=131, result=0.9999752, error=0.000310696
## n=132, result=0.9999705, error=0.0003060072
## n=133, result=0.9999659, error=0.0003014237
## n=134, result=0.9999614, error=0.0002969424
## n=135, result=0.9999571, error=0.0002925604
## n=136, result=0.9999528, error=0.0002882747
## n=137, result=0.9999486, error=0.0002840824
## n=138, result=0.9999445, error=0.000279981
## n=139, result=0.9999405, error=0.0002759677
## n=140, result=0.9999365, error=0.0002720401
## n=141, result=0.9999327, error=0.0002681958
## n=142, result=0.9999289, error=0.0002644324
## n=143, result=0.9999252, error=0.0002607477
## n=144, result=0.9999216, error=0.0002571395
## n=145, result=0.9999181, error=0.0002536057
## n=146, result=0.9999146, error=0.0002501442
## n=147, result=0.9999113, error=0.0002467531
## n=148, result=0.9999079, error=0.0002434306
## n=149, result=0.9999047, error=0.0002401747
## n=150, result=0.9999015, error=0.0002369837
## n=151, result=0.9998984, error=0.0002338558
## n=152, result=0.9998953, error=0.0002307895
## n=153, result=0.9998923, error=0.0002277832
## n=154, result=0.9998893, error=0.0002248352
## n=155, result=0.9998864, error=0.000221944
## n=156, result=0.9998836, error=0.0002191083
## n=157, result=0.9998808, error=0.0002163266
## n=158, result=0.9998781, error=0.0002135976
## n=159, result=0.9998754, error=0.0002109198
## n=160, result=0.9998728, error=0.0002082921
## n=161, result=0.9998702, error=0.0002057133
## n=162, result=0.9998677, error=0.000203182
## n=163, result=0.9998652, error=0.0002006972
## n=164, result=0.9998628, error=0.0001982577
## n=165, result=0.9998604, error=0.0001958624
## n=166, result=0.999858, error=0.0001935102
## n=167, result=0.9998557, error=0.0001912002
## n=168, result=0.9998534, error=0.0001889313
## n=169, result=0.9998512, error=0.0001867026
## n=170, result=0.999849, error=0.0001845131
## n=171, result=0.9998469, error=0.0001823618
## n=172, result=0.9998447, error=0.000180248
## n=173, result=0.9998427, error=0.0001781708
## n=174, result=0.9998406, error=0.0001761292
## n=175, result=0.9998386, error=0.0001741225
## n=176, result=0.9998366, error=0.00017215
## n=177, result=0.9998347, error=0.0001702108
## n=178, result=0.9998328, error=0.0001683042
## n=179, result=0.9998309, error=0.0001664294
## n=180, result=0.9998291, error=0.0001645858
## n=181, result=0.9998273, error=0.0001627727
## n=182, result=0.9998255, error=0.0001609893
## n=183, result=0.9998237, error=0.0001592352
## n=184, result=0.999822, error=0.0001575095
## n=185, result=0.9998203, error=0.0001558117
## n=186, result=0.9998186, error=0.0001541413
## n=187, result=0.999817, error=0.0001524976
## n=188, result=0.9998154, error=0.00015088
## n=189, result=0.9998138, error=0.0001492881
## n=190, result=0.9998122, error=0.0001477212
## n=191, result=0.9998107, error=0.0001461789
## n=192, result=0.9998092, error=0.0001446606
## n=193, result=0.9998077, error=0.0001431658
## n=194, result=0.9998062, error=0.0001416941
## n=195, result=0.9998047, error=0.000140245
## n=196, result=0.9998033, error=0.000138818
## n=197, result=0.9998019, error=0.0001374127
## n=198, result=0.9998005, error=0.0001360286
## n=199, result=0.9997992, error=0.0001346653
## n=200, result=0.9997978, error=0.0001333224
## n=201, result=0.9997965, error=0.0001319995
## n=202, result=0.9997952, error=0.0001306962
## n=203, result=0.9997939, error=0.0001294122
## n=204, result=0.9997926, error=0.0001281469
## n=205, result=0.9997914, error=0.0001269002
## n=206, result=0.9997902, error=0.0001256715
## n=207, result=0.999789, error=0.0001244606
## n=208, result=0.9997878, error=0.0001232671
## n=209, result=0.9997866, error=0.0001220908
## n=210, result=0.9997854, error=0.0001209311
## n=211, result=0.9997843, error=0.000119788
## n=212, result=0.9997832, error=0.0001186609
## n=213, result=0.999782, error=0.0001175497
## n=214, result=0.999781, error=0.0001164541
## n=215, result=0.9997799, error=0.0001153737
## n=216, result=0.9997788, error=0.0001143083
## n=217, result=0.9997778, error=0.0001132575
## n=218, result=0.9997767, error=0.0001122212
## n=219, result=0.9997757, error=0.0001111991
## n=220, result=0.9997747, error=0.0001101908
## n=221, result=0.9997737, error=0.0001091962
## n=222, result=0.9997727, error=0.0001082151
## n=223, result=0.9997717, error=0.000107247
## n=224, result=0.9997708, error=0.000106292
## n=225, result=0.9997698, error=0.0001053496
## n=226, result=0.9997689, error=0.0001044197
## n=227, result=0.999768, error=0.0001035021
## n=228, result=0.9997671, error=0.0001025965
## n=229, result=0.9997662, error=0.0001017028
## n=230, result=0.9997653, error=0.0001008207
## n=231, result=0.9997644, error=9.995e-05
## n=232, result=0.9997636, error=9.909056e-05
## n=233, result=0.9997627, error=9.824215e-05
## n=234, result=0.9997619, error=9.740461e-05
## n=235, result=0.9997611, error=9.657773e-05
## n=236, result=0.9997603, error=9.576134e-05
## n=237, result=0.9997595, error=9.495526e-05
## n=238, result=0.9997587, error=9.415932e-05
## n=239, result=0.9997579, error=9.337335e-05
## n=240, result=0.9997571, error=9.259719e-05
## n=241, result=0.9997563, error=9.183067e-05
## n=242, result=0.9997556, error=9.107362e-05
## n=243, result=0.9997548, error=9.032591e-05
## n=244, result=0.9997541, error=8.958737e-05
## n=245, result=0.9997534, error=8.885785e-05
## n=246, result=0.9997526, error=8.813722e-05
## n=247, result=0.9997519, error=8.742531e-05
## n=248, result=0.9997512, error=8.672201e-05
## n=249, result=0.9997505, error=8.602715e-05
## n=250, result=0.9997498, error=8.534062e-05
## n=251, result=0.9997492, error=8.466228e-05
## n=252, result=0.9997485, error=8.3992e-05
## n=253, result=0.9997478, error=8.332965e-05
## n=254, result=0.9997472, error=8.267511e-05
## n=255, result=0.9997465, error=8.202825e-05
## n=256, result=0.9997459, error=8.138896e-05
## n=257, result=0.9997453, error=8.075711e-05
## n=258, result=0.9997446, error=8.01326e-05
## n=259, result=0.999744, error=7.951531e-05
## n=260, result=0.9997434, error=7.890512e-05
## n=261, result=0.9997428, error=7.830194e-05
## n=262, result=0.9997422, error=7.770565e-05
## n=263, result=0.9997416, error=7.711615e-05
## n=264, result=0.999741, error=7.653333e-05
## n=265, result=0.9997405, error=7.59571e-05
## n=266, result=0.9997399, error=7.538736e-05
## n=267, result=0.9997393, error=7.482401e-05
## n=268, result=0.9997388, error=7.426695e-05
## n=269, result=0.9997382, error=7.371609e-05
## n=270, result=0.9997377, error=7.317134e-05
## n=271, result=0.9997371, error=7.263261e-05
## n=272, result=0.9997366, error=7.209981e-05
## n=273, result=0.9997361, error=7.157285e-05
## n=274, result=0.9997356, error=7.105166e-05
## n=275, result=0.999735, error=7.053614e-05
## n=276, result=0.9997345, error=7.002621e-05
## n=277, result=0.999734, error=6.952179e-05
## n=278, result=0.9997335, error=6.902281e-05
## n=279, result=0.999733, error=6.852918e-05
## n=280, result=0.9997325, error=6.804084e-05
## n=281, result=0.9997321, error=6.755769e-05
## n=282, result=0.9997316, error=6.707968e-05
## n=283, result=0.9997311, error=6.660673e-05
## n=284, result=0.9997306, error=6.613876e-05
## n=285, result=0.9997302, error=6.567571e-05
## n=286, result=0.9997297, error=6.521751e-05
## n=287, result=0.9997293, error=6.476409e-05
## n=288, result=0.9997288, error=6.431539e-05
## n=289, result=0.9997284, error=6.387133e-05
## n=290, result=0.9997279, error=6.343186e-05
## n=291, result=0.9997275, error=6.299691e-05
## n=292, result=0.9997271, error=6.256643e-05
## n=293, result=0.9997266, error=6.214034e-05
## n=294, result=0.9997262, error=6.17186e-05
## n=295, result=0.9997258, error=6.130113e-05
## n=296, result=0.9997254, error=6.088789e-05
## n=297, result=0.999725, error=6.047882e-05
## n=298, result=0.9997246, error=6.007386e-05
## n=299, result=0.9997242, error=5.967295e-05
## n=300, result=0.9997238, error=5.927605e-05
## n=301, result=0.9997234, error=5.88831e-05
## n=302, result=0.999723, error=5.849404e-05
## n=303, result=0.9997226, error=5.810883e-05
## n=304, result=0.9997222, error=5.772741e-05
## n=305, result=0.9997218, error=5.734974e-05
## n=306, result=0.9997215, error=5.697577e-05
## n=307, result=0.9997211, error=5.660544e-05
## n=308, result=0.9997207, error=5.623872e-05
## n=309, result=0.9997204, error=5.587555e-05
## n=310, result=0.99972, error=5.551588e-05
## n=311, result=0.9997197, error=5.515969e-05
## n=312, result=0.9997193, error=5.480691e-05
## n=313, result=0.999719, error=5.445751e-05
## n=314, result=0.9997186, error=5.411144e-05
## n=315, result=0.9997183, error=5.376866e-05
## n=316, result=0.9997179, error=5.342913e-05
## n=317, result=0.9997176, error=5.309281e-05
## n=318, result=0.9997173, error=5.275965e-05
## n=319, result=0.9997169, error=5.242963e-05
## n=320, result=0.9997166, error=5.210269e-05
## n=321, result=0.9997163, error=5.177881e-05
## n=322, result=0.999716, error=5.145793e-05
## n=323, result=0.9997156, error=5.114003e-05
## n=324, result=0.9997153, error=5.082508e-05
## n=325, result=0.999715, error=5.051302e-05
## n=326, result=0.9997147, error=5.020383e-05
## n=327, result=0.9997144, error=4.989748e-05
## n=328, result=0.9997141, error=4.959392e-05
## n=329, result=0.9997138, error=4.929312e-05
## n=330, result=0.9997135, error=4.899506e-05
## n=331, result=0.9997132, error=4.869969e-05
## n=332, result=0.9997129, error=4.840699e-05
## n=333, result=0.9997126, error=4.811692e-05
## n=334, result=0.9997123, error=4.782945e-05
## n=335, result=0.999712, error=4.754455e-05
## n=336, result=0.9997118, error=4.72622e-05
## n=337, result=0.9997115, error=4.698235e-05
## n=338, result=0.9997112, error=4.670498e-05
## n=339, result=0.9997109, error=4.643007e-05
## n=340, result=0.9997107, error=4.615757e-05
## n=341, result=0.9997104, error=4.588747e-05
## n=342, result=0.9997101, error=4.561974e-05
## n=343, result=0.9997099, error=4.535434e-05
## n=344, result=0.9997096, error=4.509125e-05
## n=345, result=0.9997093, error=4.483045e-05
## n=346, result=0.9997091, error=4.457191e-05
## n=347, result=0.9997088, error=4.43156e-05
## n=348, result=0.9997086, error=4.406149e-05
## n=349, result=0.9997083, error=4.380957e-05
## n=350, result=0.9997081, error=4.35598e-05
## n=351, result=0.9997078, error=4.331217e-05
## n=352, result=0.9997076, error=4.306664e-05
## n=353, result=0.9997073, error=4.282319e-05
## n=354, result=0.9997071, error=4.258181e-05
## n=355, result=0.9997068, error=4.234246e-05
## n=356, result=0.9997066, error=4.210513e-05
## n=357, result=0.9997064, error=4.186979e-05
## n=358, result=0.9997061, error=4.163642e-05
## n=359, result=0.9997059, error=4.140499e-05
## n=360, result=0.9997057, error=4.117549e-05
## n=361, result=0.9997054, error=4.09479e-05
## n=362, result=0.9997052, error=4.072219e-05
## n=363, result=0.999705, error=4.049834e-05
## n=364, result=0.9997048, error=4.027633e-05
## n=365, result=0.9997046, error=4.005615e-05
## n=366, result=0.9997043, error=3.983777e-05
## n=367, result=0.9997041, error=3.962117e-05
## n=368, result=0.9997039, error=3.940633e-05
## n=369, result=0.9997037, error=3.919324e-05
## n=370, result=0.9997035, error=3.898188e-05
## n=371, result=0.9997033, error=3.877222e-05
## n=372, result=0.9997031, error=3.856425e-05
## n=373, result=0.9997029, error=3.835795e-05
## n=374, result=0.9997027, error=3.81533e-05
## n=375, result=0.9997025, error=3.795029e-05
## n=376, result=0.9997022, error=3.774889e-05
## n=377, result=0.999702, error=3.75491e-05
## n=378, result=0.9997019, error=3.735089e-05
## n=379, result=0.9997017, error=3.715424e-05
## n=380, result=0.9997015, error=3.695915e-05
## n=381, result=0.9997013, error=3.676559e-05
## n=382, result=0.9997011, error=3.657355e-05
## n=383, result=0.9997009, error=3.638301e-05
## n=384, result=0.9997007, error=3.619396e-05
## n=385, result=0.9997005, error=3.600638e-05
## n=386, result=0.9997003, error=3.582025e-05
## n=387, result=0.9997001, error=3.563557e-05
## n=388, result=0.9997, error=3.545231e-05
## n=389, result=0.9996998, error=3.527046e-05
## n=390, result=0.9996996, error=3.509001e-05
## n=391, result=0.9996994, error=3.491095e-05
## n=392, result=0.9996992, error=3.473325e-05
## n=393, result=0.9996991, error=3.455691e-05
## n=394, result=0.9996989, error=3.43819e-05
## n=395, result=0.9996987, error=3.420823e-05
## n=396, result=0.9996985, error=3.403587e-05
## n=397, result=0.9996984, error=3.386481e-05
## n=398, result=0.9996982, error=3.369503e-05
## n=399, result=0.999698, error=3.352654e-05
## n=400, result=0.9996979, error=3.33593e-05
## n=401, result=0.9996977, error=3.319332e-05
## n=402, result=0.9996975, error=3.302857e-05
## n=403, result=0.9996974, error=3.286504e-05
## n=404, result=0.9996972, error=3.270273e-05
## n=405, result=0.999697, error=3.254162e-05
## n=406, result=0.9996969, error=3.23817e-05
## n=407, result=0.9996967, error=3.222296e-05
## n=408, result=0.9996966, error=3.206538e-05
## n=409, result=0.9996964, error=3.190895e-05
## n=410, result=0.9996963, error=3.175367e-05
## n=411, result=0.9996961, error=3.159953e-05
## n=412, result=0.9996959, error=3.14465e-05
## n=413, result=0.9996958, error=3.129458e-05
## n=414, result=0.9996956, error=3.114376e-05
## n=415, result=0.9996955, error=3.099403e-05
## n=416, result=0.9996953, error=3.084538e-05
## n=417, result=0.9996952, error=3.06978e-05
## n=418, result=0.9996951, error=3.055128e-05
## n=419, result=0.9996949, error=3.04058e-05
## n=420, result=0.9996948, error=3.026136e-05
## n=421, result=0.9996946, error=3.011795e-05
## n=422, result=0.9996945, error=2.997556e-05
## n=423, result=0.9996943, error=2.983418e-05
## n=424, result=0.9996942, error=2.969379e-05
## n=425, result=0.9996941, error=2.95544e-05
## n=426, result=0.9996939, error=2.941598e-05
## n=427, result=0.9996938, error=2.927854e-05
## n=428, result=0.9996936, error=2.914206e-05
## n=429, result=0.9996935, error=2.900653e-05
## n=430, result=0.9996934, error=2.887195e-05
## n=431, result=0.9996932, error=2.87383e-05
## n=432, result=0.9996931, error=2.860558e-05
## n=433, result=0.999693, error=2.847378e-05
## n=434, result=0.9996928, error=2.834289e-05
## n=435, result=0.9996927, error=2.82129e-05
## n=436, result=0.9996926, error=2.80838e-05
## n=437, result=0.9996925, error=2.795559e-05
## n=438, result=0.9996923, error=2.782826e-05
## n=439, result=0.9996922, error=2.770179e-05
## n=440, result=0.9996921, error=2.757619e-05
## n=441, result=0.999692, error=2.745144e-05
## n=442, result=0.9996918, error=2.732753e-05
## n=443, result=0.9996917, error=2.720447e-05
## n=444, result=0.9996916, error=2.708223e-05
## n=445, result=0.9996915, error=2.696082e-05
## n=446, result=0.9996913, error=2.684022e-05
## n=447, result=0.9996912, error=2.672043e-05
## n=448, result=0.9996911, error=2.660145e-05
## n=449, result=0.999691, error=2.648325e-05
## n=450, result=0.9996909, error=2.636585e-05
## n=451, result=0.9996907, error=2.624922e-05
## n=452, result=0.9996906, error=2.613337e-05
## n=453, result=0.9996905, error=2.601828e-05
## n=454, result=0.9996904, error=2.590396e-05
## n=455, result=0.9996903, error=2.579038e-05
## n=456, result=0.9996902, error=2.567755e-05
## n=457, result=0.9996901, error=2.556547e-05
## n=458, result=0.99969, error=2.545411e-05
## n=459, result=0.9996898, error=2.534349e-05
## n=460, result=0.9996897, error=2.523358e-05
## n=461, result=0.9996896, error=2.512439e-05
## n=462, result=0.9996895, error=2.50159e-05
## n=463, result=0.9996894, error=2.490812e-05
## n=464, result=0.9996893, error=2.480104e-05
## n=465, result=0.9996892, error=2.469464e-05
## n=466, result=0.9996891, error=2.458893e-05
## n=467, result=0.999689, error=2.44839e-05
## n=468, result=0.9996889, error=2.437954e-05
## n=469, result=0.9996888, error=2.427584e-05
## n=470, result=0.9996887, error=2.417281e-05
## n=471, result=0.9996886, error=2.407043e-05
## n=472, result=0.9996885, error=2.396871e-05
## n=473, result=0.9996884, error=2.386763e-05
## n=474, result=0.9996883, error=2.376718e-05
## n=475, result=0.9996882, error=2.366737e-05
## n=476, result=0.9996881, error=2.356819e-05
## n=477, result=0.999688, error=2.346963e-05
## n=478, result=0.9996879, error=2.337169e-05
## n=479, result=0.9996878, error=2.327437e-05
## n=480, result=0.9996877, error=2.317765e-05
## n=481, result=0.9996876, error=2.308153e-05
## n=482, result=0.9996875, error=2.298601e-05
## n=483, result=0.9996874, error=2.289108e-05
## n=484, result=0.9996873, error=2.279675e-05
## n=485, result=0.9996872, error=2.270299e-05
## n=486, result=0.9996871, error=2.260981e-05
## n=487, result=0.999687, error=2.251721e-05
## n=488, result=0.9996869, error=2.242517e-05
## n=489, result=0.9996868, error=2.23337e-05
## n=490, result=0.9996867, error=2.224279e-05
## n=491, result=0.9996867, error=2.215243e-05
## n=492, result=0.9996866, error=2.206262e-05
## n=493, result=0.9996865, error=2.197336e-05
## n=494, result=0.9996864, error=2.188464e-05
## n=495, result=0.9996863, error=2.179646e-05
## n=496, result=0.9996862, error=2.170881e-05
## n=497, result=0.9996861, error=2.162169e-05
## n=498, result=0.999686, error=2.153509e-05
## n=499, result=0.9996859, error=2.144902e-05
## n=500, result=0.9996859, error=2.136346e-05
## n=501, result=0.9996858, error=2.127841e-05
## n=502, result=0.9996857, error=2.119387e-05
## n=503, result=0.9996856, error=2.110983e-05
## n=504, result=0.9996855, error=2.102629e-05
## n=505, result=0.9996854, error=2.094325e-05
## n=506, result=0.9996854, error=2.08607e-05
## n=507, result=0.9996853, error=2.077864e-05
## n=508, result=0.9996852, error=2.069706e-05
## n=509, result=0.9996851, error=2.061596e-05
## n=510, result=0.999685, error=2.053534e-05
## n=511, result=0.999685, error=2.04552e-05
## n=512, result=0.9996849, error=2.037552e-05
## n=513, result=0.9996848, error=2.02963e-05
## n=514, result=0.9996847, error=2.021755e-05
## n=515, result=0.9996846, error=2.013926e-05
## n=516, result=0.9996846, error=2.006142e-05
## n=517, result=0.9996845, error=1.998403e-05
## n=518, result=0.9996844, error=1.990709e-05
## n=519, result=0.9996843, error=1.98306e-05
## n=520, result=0.9996843, error=1.975454e-05
## n=521, result=0.9996842, error=1.967893e-05
## n=522, result=0.9996841, error=1.960374e-05
## n=523, result=0.999684, error=1.952899e-05
## n=524, result=0.999684, error=1.945467e-05
## n=525, result=0.9996839, error=1.938077e-05
## n=526, result=0.9996838, error=1.930729e-05
## n=527, result=0.9996837, error=1.923423e-05
## n=528, result=0.9996837, error=1.916158e-05
## n=529, result=0.9996836, error=1.908935e-05
## n=530, result=0.9996835, error=1.901752e-05
## n=531, result=0.9996834, error=1.89461e-05
## n=532, result=0.9996834, error=1.887508e-05
## n=533, result=0.9996833, error=1.880446e-05
## n=534, result=0.9996832, error=1.873424e-05
## n=535, result=0.9996832, error=1.866441e-05
## n=536, result=0.9996831, error=1.859497e-05
## n=537, result=0.999683, error=1.852592e-05
## n=538, result=0.999683, error=1.845725e-05
## n=539, result=0.9996829, error=1.838897e-05
## n=540, result=0.9996828, error=1.832106e-05
## n=541, result=0.9996828, error=1.825353e-05
## n=542, result=0.9996827, error=1.818638e-05
## n=543, result=0.9996826, error=1.811959e-05
## n=544, result=0.9996826, error=1.805318e-05
## n=545, result=0.9996825, error=1.798712e-05
## n=546, result=0.9996824, error=1.792143e-05
## n=547, result=0.9996824, error=1.78561e-05
## n=548, result=0.9996823, error=1.779113e-05
## n=549, result=0.9996822, error=1.772651e-05
## n=550, result=0.9996822, error=1.766225e-05
## n=551, result=0.9996821, error=1.759833e-05
## n=552, result=0.999682, error=1.753476e-05
## n=553, result=0.999682, error=1.747154e-05
## n=554, result=0.9996819, error=1.740866e-05
## n=555, result=0.9996818, error=1.734612e-05
## n=556, result=0.9996818, error=1.728391e-05
## n=557, result=0.9996817, error=1.722204e-05
## n=558, result=0.9996817, error=1.71605e-05
## n=559, result=0.9996816, error=1.709929e-05
## n=560, result=0.9996815, error=1.703841e-05
## n=561, result=0.9996815, error=1.697786e-05
## n=562, result=0.9996814, error=1.691762e-05
## n=563, result=0.9996814, error=1.685771e-05
## n=564, result=0.9996813, error=1.679812e-05
## n=565, result=0.9996812, error=1.673884e-05
## n=566, result=0.9996812, error=1.667988e-05
## n=567, result=0.9996811, error=1.662123e-05
## n=568, result=0.9996811, error=1.656288e-05
## n=569, result=0.999681, error=1.650485e-05
## n=570, result=0.9996809, error=1.644712e-05
## n=571, result=0.9996809, error=1.638969e-05
## n=572, result=0.9996808, error=1.633257e-05
## n=573, result=0.9996808, error=1.627574e-05
## n=574, result=0.9996807, error=1.621921e-05
## n=575, result=0.9996807, error=1.616297e-05
## n=576, result=0.9996806, error=1.610703e-05
## n=577, result=0.9996806, error=1.605138e-05
## n=578, result=0.9996805, error=1.599601e-05
## n=579, result=0.9996804, error=1.594094e-05
## n=580, result=0.9996804, error=1.588615e-05
## n=581, result=0.9996803, error=1.583164e-05
## n=582, result=0.9996803, error=1.577741e-05
## n=583, result=0.9996802, error=1.572346e-05
## n=584, result=0.9996802, error=1.566978e-05
## n=585, result=0.9996801, error=1.561638e-05
## n=586, result=0.9996801, error=1.556326e-05
## n=587, result=0.99968, error=1.551041e-05
## n=588, result=0.99968, error=1.545782e-05
## n=589, result=0.9996799, error=1.54055e-05
## n=590, result=0.9996799, error=1.535345e-05
## n=591, result=0.9996798, error=1.530167e-05
## n=592, result=0.9996798, error=1.525014e-05
## n=593, result=0.9996797, error=1.519888e-05
## n=594, result=0.9996796, error=1.514787e-05
## n=595, result=0.9996796, error=1.509712e-05
## n=596, result=0.9996795, error=1.504663e-05
## n=597, result=0.9996795, error=1.499639e-05
## n=598, result=0.9996794, error=1.49464e-05
## n=599, result=0.9996794, error=1.489666e-05
## n=600, result=0.9996793, error=1.484717e-05
## n=601, result=0.9996793, error=1.479793e-05
## n=602, result=0.9996792, error=1.474893e-05
## n=603, result=0.9996792, error=1.470018e-05
## n=604, result=0.9996792, error=1.465167e-05
## n=605, result=0.9996791, error=1.46034e-05
## n=606, result=0.9996791, error=1.455536e-05
## n=607, result=0.999679, error=1.450757e-05
## n=608, result=0.999679, error=1.446001e-05
## n=609, result=0.9996789, error=1.441268e-05
## n=610, result=0.9996789, error=1.436559e-05
## n=611, result=0.9996788, error=1.431873e-05
## n=612, result=0.9996788, error=1.427209e-05
## n=613, result=0.9996787, error=1.422569e-05
## n=614, result=0.9996787, error=1.417951e-05
## n=615, result=0.9996786, error=1.413356e-05
## n=616, result=0.9996786, error=1.408783e-05
## n=617, result=0.9996785, error=1.404232e-05
## n=618, result=0.9996785, error=1.399703e-05
## n=619, result=0.9996785, error=1.395197e-05
## n=620, result=0.9996784, error=1.390712e-05
## n=621, result=0.9996784, error=1.386248e-05
## n=622, result=0.9996783, error=1.381806e-05
## n=623, result=0.9996783, error=1.377386e-05
## n=624, result=0.9996782, error=1.372987e-05
## n=625, result=0.9996782, error=1.368609e-05
## n=626, result=0.9996781, error=1.364252e-05
## n=627, result=0.9996781, error=1.359915e-05
## n=628, result=0.9996781, error=1.3556e-05
## n=629, result=0.999678, error=1.351305e-05
## n=630, result=0.999678, error=1.34703e-05
## n=631, result=0.9996779, error=1.342776e-05
## n=632, result=0.9996779, error=1.338542e-05
## n=633, result=0.9996778, error=1.334328e-05
## n=634, result=0.9996778, error=1.330134e-05
## n=635, result=0.9996778, error=1.325959e-05
## n=636, result=0.9996777, error=1.321805e-05
## n=637, result=0.9996777, error=1.31767e-05
## n=638, result=0.9996776, error=1.313554e-05
## n=639, result=0.9996776, error=1.309458e-05
## n=640, result=0.9996776, error=1.30538e-05
## n=641, result=0.9996775, error=1.301322e-05
## n=642, result=0.9996775, error=1.297283e-05
## n=643, result=0.9996774, error=1.293263e-05
## n=644, result=0.9996774, error=1.289261e-05
## n=645, result=0.9996774, error=1.285278e-05
## n=646, result=0.9996773, error=1.281314e-05
## n=647, result=0.9996773, error=1.277367e-05
## n=648, result=0.9996772, error=1.273439e-05
## n=649, result=0.9996772, error=1.26953e-05
## n=650, result=0.9996772, error=1.265638e-05
## n=651, result=0.9996771, error=1.261764e-05
## n=652, result=0.9996771, error=1.257908e-05
## n=653, result=0.999677, error=1.25407e-05
## n=654, result=0.999677, error=1.250249e-05
## n=655, result=0.999677, error=1.246446e-05
## n=656, result=0.9996769, error=1.24266e-05
## n=657, result=0.9996769, error=1.238891e-05
## n=658, result=0.9996769, error=1.23514e-05
## n=659, result=0.9996768, error=1.231406e-05
## n=660, result=0.9996768, error=1.227688e-05
## n=661, result=0.9996767, error=1.223988e-05
## n=662, result=0.9996767, error=1.220304e-05
## n=663, result=0.9996767, error=1.216637e-05
## n=664, result=0.9996766, error=1.212986e-05
## n=665, result=0.9996766, error=1.209352e-05
## n=666, result=0.9996766, error=1.205735e-05
## n=667, result=0.9996765, error=1.202133e-05
## n=668, result=0.9996765, error=1.198548e-05
## n=669, result=0.9996764, error=1.194979e-05
## n=670, result=0.9996764, error=1.191425e-05
## n=671, result=0.9996764, error=1.187888e-05
## n=672, result=0.9996763, error=1.184366e-05
## n=673, result=0.9996763, error=1.18086e-05
## n=674, result=0.9996763, error=1.17737e-05
## n=675, result=0.9996762, error=1.173895e-05
## n=676, result=0.9996762, error=1.170436e-05
## n=677, result=0.9996762, error=1.166992e-05
## n=678, result=0.9996761, error=1.163563e-05
## n=679, result=0.9996761, error=1.160149e-05
## n=680, result=0.9996761, error=1.15675e-05
## n=681, result=0.999676, error=1.153366e-05
## n=682, result=0.999676, error=1.149998e-05
## n=683, result=0.999676, error=1.146643e-05
## n=684, result=0.9996759, error=1.143304e-05
## n=685, result=0.9996759, error=1.139979e-05
## n=686, result=0.9996759, error=1.136669e-05
## n=687, result=0.9996758, error=1.133373e-05
## n=688, result=0.9996758, error=1.130092e-05
## n=689, result=0.9996758, error=1.126825e-05
## n=690, result=0.9996757, error=1.123572e-05
## n=691, result=0.9996757, error=1.120333e-05
## n=692, result=0.9996757, error=1.117108e-05
## n=693, result=0.9996756, error=1.113897e-05
## n=694, result=0.9996756, error=1.1107e-05
## n=695, result=0.9996756, error=1.107517e-05
## n=696, result=0.9996755, error=1.104348e-05
## n=697, result=0.9996755, error=1.101192e-05
## n=698, result=0.9996755, error=1.098049e-05
## n=699, result=0.9996754, error=1.094921e-05
## n=700, result=0.9996754, error=1.091805e-05
## n=701, result=0.9996754, error=1.088703e-05
## n=702, result=0.9996754, error=1.085614e-05
## n=703, result=0.9996753, error=1.082538e-05
## n=704, result=0.9996753, error=1.079476e-05
## n=705, result=0.9996753, error=1.076426e-05
## n=706, result=0.9996752, error=1.07339e-05
## n=707, result=0.9996752, error=1.070366e-05
## n=708, result=0.9996752, error=1.067355e-05
## n=709, result=0.9996751, error=1.064357e-05
## n=710, result=0.9996751, error=1.061371e-05
## n=711, result=0.9996751, error=1.058398e-05
## n=712, result=0.9996751, error=1.055438e-05
## n=713, result=0.999675, error=1.05249e-05
## n=714, result=0.999675, error=1.049554e-05
## n=715, result=0.999675, error=1.046631e-05
## n=716, result=0.9996749, error=1.04372e-05
## n=717, result=0.9996749, error=1.040821e-05
## n=718, result=0.9996749, error=1.037934e-05
## n=719, result=0.9996749, error=1.035059e-05
## n=720, result=0.9996748, error=1.032197e-05
## n=721, result=0.9996748, error=1.029346e-05
## n=722, result=0.9996748, error=1.026507e-05
## n=723, result=0.9996747, error=1.023679e-05
## n=724, result=0.9996747, error=1.020864e-05
## n=725, result=0.9996747, error=1.01806e-05
## n=726, result=0.9996747, error=1.015268e-05
## n=727, result=0.9996746, error=1.012487e-05
## n=728, result=0.9996746, error=1.009717e-05
## n=729, result=0.9996746, error=1.006959e-05
## n=730, result=0.9996745, error=1.004213e-05
## n=731, result=0.9996745, error=1.001477e-05
## n=732, result=0.9996745, error=9.98753e-06

untuk metode Trapezoida diperoleh nilai CDF yaitu 1.064635, dengan dimasukan nilai toleransi yaitu 0,00001 dan iterasi sebanyak 1000, nilai dengan metode ini mendekati nilai exact nya dengan erro yang juga semakin mengecil.

Metode simpson

simpson_n(f1,0,4,n = 4)

## [1] 1.058815

exact_value=0.9996645
tol <-  0.00001
err <- 1
n = 9

while(err>tol){
  res_simp <- simpson_n(f1,0,4,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=9, result=1.00462, error=0.004955555
## n=10, result=1.001777, error=0.002112444
## n=11, result=1.001777, error=0.002112444
## n=12, result=1.000706, error=0.00104161
## n=13, result=1.000706, error=0.00104161
## n=14, result=1.000234, error=0.0005699298
## n=15, result=1.000234, error=0.0005699298
## n=16, result=1.000002, error=0.000337077
## n=17, result=1.000002, error=0.000337077
## n=18, result=0.9998762, error=0.0002117416
## n=19, result=0.9998762, error=0.0002117416
## n=20, result=0.999804, error=0.0001395486
## n=21, result=0.999804, error=0.0001395486
## n=22, result=0.9997601, error=9.563677e-05
## n=23, result=0.9997601, error=9.563677e-05
## n=24, result=0.9997322, error=6.7705e-05
## n=25, result=0.9997322, error=6.7705e-05
## n=26, result=0.9997138, error=4.926109e-05
## n=27, result=0.9997138, error=4.926109e-05
## n=28, result=0.9997012, error=3.668996e-05
## n=29, result=0.9997012, error=3.668996e-05
## n=30, result=0.9996924, error=2.788524e-05
## n=31, result=0.9996924, error=2.788524e-05
## n=32, result=0.9996861, error=2.157112e-05
## n=33, result=0.9996861, error=2.157112e-05
## n=34, result=0.9996814, error=1.694843e-05
## n=35, result=0.9996814, error=1.694843e-05
## n=36, result=0.999678, error=1.350169e-05
## n=37, result=0.999678, error=1.350169e-05
## n=38, result=0.9996754, error=1.088964e-05
## n=39, result=0.9996754, error=1.088964e-05
## n=40, result=0.9996734, error=8.881142e-06

dengan metode simpson diperoleh nilai yaitu 1.058815, dengan toleransi 0,00001 maka hasil iterasi semakin mendekati nilai exact dari CDF.

Metode n-point gaussian

Sebelum kita menggunakan nilai-nilai koefisien dan titik gauss, perlu dilakukan transformasi sehingga domain integral menjadi [−1,1].

\(x= \frac {1}{2}[t(b-a)+a+b]=\frac {1}{2} [t(4-0)+4+0]=\frac {4t+4}{2} = 2(t+1)\)

dan

\(dx= \frac {1}{2}(b-a)dt = \frac{1}{2} (4-0) dt = 2 dt\)

sehingga diperoleh intergralnya dengan batas [-1,1]

\(\int_{0}^{1}2exp^{-2x}dx =\int_{-1}^{1} 2exp^{-2(2t+2)} 2dt=\int_{-1}^{1} 4exp^{-4t-4} dt\) \(\approx I= C_1 f(t_1)+C_2 f(t_2) + C_3 f(t_3)+C_4 f(t_4)\)

dengan batas [-1,1] maka nilai dengan R diperoleh adalah sebagai berikut;

fx <- function(x){
  4*exp(-4*x-4)
}
gL <- gaussLegendre(n = 9,a = -1,1)
gL

## $x
## [1] -9.681602e-01 -8.360311e-01 -6.133714e-01 -3.242534e-01 -2.220446e-16
## [6]  3.242534e-01  6.133714e-01  8.360311e-01  9.681602e-01
## 
## $w
## [1] 0.08127439 0.18064816 0.26061070 0.31234708 0.33023936 0.31234708 0.26061070
## [8] 0.18064816 0.08127439

Ci <- gL$w # koefisien
Ci

## [1] 0.08127439 0.18064816 0.26061070 0.31234708 0.33023936 0.31234708 0.26061070
## [8] 0.18064816 0.08127439

xi <- gL$x # gauss point
xi

## [1] -9.681602e-01 -8.360311e-01 -6.133714e-01 -3.242534e-01 -2.220446e-16
## [6]  3.242534e-01  6.133714e-01  8.360311e-01  9.681602e-01

I <- sum(Ci * fx(xi))
I

## [1] 0.9996645

jika batas asli a=0,b=4 sebagai berikut ;

gL <- gaussLegendre(n = 9,a = 0,4)
Ci <- gL$w # koefisien
Ci

## [1] 0.1625488 0.3612963 0.5212214 0.6246942 0.6604787 0.6246942 0.5212214
## [8] 0.3612963 0.1625488

xi <- gL$x # gauss point
xi

## [1] 0.06367952 0.32793779 0.77325713 1.35149315 2.00000000 2.64850685 3.22674287
## [8] 3.67206221 3.93632048

I <- sum(Ci * f1(xi))
I

## [1] 0.9996645

exact_value=0.9996645
tol <-  0.00001
err <- 1
n = 9

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = 0,4)
  Ci <- gL$w # koefisien
 xi <- gL$x # gauss point
  res_gl <- sum(Ci * f1(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=9, result=0.9996645, error=3.73609e-08

dengan menggunakan metode n-point gaussian dan dipilih n=9 maka hasil yang diperoleh sama dengan nilai exact dari CDF nya, dan dengan toleransi 0,00001.

Metode Monte Carlo

diketahui fungsi PDF dari suatu fungsi exponensial sebagai berikut :

\(f(x=4| \lambda = 2)=\lambda exp-\lambda x\) , dimana \(x \epsilon (0,\infty)\)

\(f(x=4| \lambda = 2)=2 exp-2x\)

maka, CDF dari suatu fungsi yaitu

\(F(x)=\int_{-\infty}^\infty f(t)dt\)

sehingga CDF dari fungsi \(f(x)\) diatas yaitu

\(F(x=4| \lambda = 2)=\int_{0}^4 2 exp-2t dt\)

dengan nilai \(x=4\) maka batas atas dan bawah adalah keduanya positif sehingga dapat di lakukan perhitungan nilai CDF dengan metode monte carlo, sebagai berikut :

x <- 4
Fm <- function(x){
  2 * exp(-2*x)
}

mc_integral <- function(ftn, a, b,m){
  #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)
}

set.seed(8)
int_g <- mc_integral(Fm,0, 4,m = 2114)
int_g

## [1] 0.9993944

untuk metode monte carlo ddiperoleh nilai yaitu 0,9993944 yang mendekati nilai exactnya, digunakan trial and error berapa m yang digunakan sehingga benar-benar mendekati nilai exact CDF dan diperoleh nilai m=2114 yang paling mendekati menggunakan metode monte carlo ini.

Kesimpulan

Dari 4 metode yang digunakan untuk soal nomor 2 untuk mencari nilai CDF yang paling mendekati dengan nilai exact yaitu 0.9996645, maka di pilih metode terbaik dengan nilai yang sama yaitu metode gauss quadratur dengan n=9 dan toleransi 0,00001 yang digunakan. Diperoleh bahwa berdasarkan soal 2 semakin besar n yang digunakan maka nilai semakin mendekati nilai exact dari CDF.