Pembuktian Teorema Limit Pusat pada Sebaran Normal, Eksponensial, Seragam, dan Binomial

library(rmdformats)

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

set.seed(1902)
k<-1000 # ulangan

Normal

n<-5 #banyak sampel
x11<-matrix(rnorm(n*k),k) #bangkitkan sampel
x11<-apply(x11,1,mean) #simpan rataan
hist(x11, main = "Sebaran Percontohan Normal 5 Sampel"); mean(x11); var(x11)

## [1] 0.01487933

## [1] 0.2154885

n<-50 #banyak sampel
x12<-matrix(rnorm(n*k),k) #bangkitkan sampel
x12<-apply(x12,1,mean) #simpan rataan
hist(x12, main = "Sebaran Percontohan Normal 50 Sampel"); mean(x12); var(x12)

## [1] 0.001371539

## [1] 0.01986017

n<-500 #banyak sampel
x13<-matrix(rnorm(n*k),k) #bangkitkan sampel
x13<-apply(x13,1,mean) #simpan rataan
hist(x13, main = "Sebaran Percontohan Normal 500 Sampel"); mean(x13); var(x13)

## [1] 7.247556e-05

## [1] 0.001780417

Eksponensial

n<-5 #banyak sampel
x21<-matrix(rexp(n*k),k) #bangkitkan sampel
x21<-apply(x21,1,mean) #simpan rataan
hist(x21, main = "Sebaran Percontohan Eksponensial 5 Sampel"); mean(x21); var(x21)

## [1] 0.9924758

## [1] 0.196669

n<-50 #banyak sampel
x22<-matrix(rexp(n*k),k) #bangkitkan sampel
x22<-apply(x22,1,mean) #simpan rataan
hist(x22, main = "Sebaran Percontohan Eksponensial 50 Sampel") ; mean(x22); var(x22)

## [1] 1.008955

## [1] 0.02085168

n<-500 #banyak sampel
x23<-matrix(rexp(n*k),k) #bangkitkan sampel
x23<-apply(x23,1,mean) #simpan rataan
hist(x23, main = "Sebaran Percontohan Eksponensial 500 Sampel"); mean(x23); var(x23)

## [1] 0.9972613

## [1] 0.001893484

Seragam

n<-5 #banyak sampel
x31<-matrix(runif(n*k),k) #bangkitkan sampel
x31<-apply(x31,1,mean) #simpan rataan
hist(x31, main = "Sebaran Percontohan Seragam 5 Sampel"); mean(x31); var(x31)

## [1] 0.4985274

## [1] 0.01709433

n<-50 #banyak sampel
x32<-matrix(runif(n*k),k) #bangkitkan sampel
x32<-apply(x32,1,mean) #simpan rataan
hist(x32, main = "Sebaran Percontohan Seragam 50 Sampel"); mean(x32); var(x32)

## [1] 0.5010423

## [1] 0.001609224

n<-500 #banyak sampel
x33<-matrix(runif(n*k),k) #bangkitkan sampel
x33<-apply(x33,1,mean) #simpan rataan
hist(x33, main = "Sebaran Percontohan Seragam 500 Sampel"); mean(x33); var(x33)

## [1] 0.5005761

## [1] 0.0001618947

Binomial

n_trials <- 5  # Jumlah uji percobaan
prob_success <- 0.9  # Probabilitas sukses


n<-5 #banyak sampel
x41<-matrix(rbinom(n*k, size = n_trials, prob = prob_success),k) #bangkitkan sampel
x41<-apply(x41,1,mean) #simpan rataan
hist(x41, main = "Sebaran Percontohan Binomial 5 Sampel"); mean(x41); var(x41)

## [1] 4.4894

## [1] 0.09197962

n_trials <- 5  # Jumlah uji percobaan
prob_success <- 0.9  # Probabilitas sukses


n<-50 #banyak sampel
x42<-matrix(rbinom(n*k, size = n_trials, prob = prob_success),k) #bangkitkan sampel
x42<-apply(x42,1,mean) #simpan rataan
hist(x42, main = "Sebaran Percontohan Binomial 50 Sampel"); mean(x42); var(x42)

## [1] 4.49868

## [1] 0.009204262

n_trials <- 5  # Jumlah uji percobaan
prob_success <- 0.9  # Probabilitas sukses


n<-500 #banyak sampel
x43<-matrix(rbinom(n*k, size = n_trials, prob = prob_success),k) #bangkitkan sampel
x43<-apply(x43,1,mean) #simpan rataan
hist(x43, main = "Sebaran Percontohan Binomial 500 Sampel"); mean(x43); var(x43)

## [1] 4.50107

## [1] 0.0008935126

Perbandingan

par(mfrow=c(4,3))
hist(x11, main = "Normal 5 Sampel",xlim = c(-2,2));hist(x12, main = "Normal 50 Sampel",xlim = c(-2,2));hist(x13, main = "Normal 500 Sampel",xlim = c(-2,2));
hist(x21, main = "Eksponensial 5 Sampel",xlim = c(0,3));hist(x22, main = "Eksponensial 50 Sampel",xlim = c(0,3));hist(x23, main = "Eksponensial 500 Sampel",xlim = c(0,3));
hist(x31, main = "Seragam 5 Sampel",xlim = c(0,1));hist(x32, main = "Seragam 50 Sampel",xlim = c(0,1));hist(x33, main = "Seragam 500 Sampel",xlim = c(0,1))
hist(x41, main = "Binomial 5 Sampel",xlim = c(0,5));hist(x22, main = "Binomial 50 Sampel",xlim = c(0,5));hist(x23, main = "Binomial 500 Sampel",xlim = c(0,5));

Kesimpulan

Semakin besar jumlah sampel yang digunakan, maka rataan sampel semakin mendekati sebatan normal. Hal ini sesuai dengan dalil teorema limit pusat. Hal ini terlihat ketika n bernilai 5, rataan sebaram eksponensial, seragam, dan binomial tidak menyebar secara normal. Dimana eksponensial menjulur ke kanan, seragam melandai, dan binomial menjulur ke kiri. Ketika sampel dinaikkan menjadi 50, semua rataan sebaran cenderung menyebar normal. Terutama ketika sampel berjumlah 500.