Populasi Tak Terhingga

Normal

  1. n=10
k<-1000
n<-10
x11<-matrix(rnorm(n*k),k)
x11<-apply(x11,1,mean)
hist(x11); mean(x11); var(x11)

## [1] -0.00947418
## [1] 0.1087172
  1. n=30
k<-1000
n<-30
x12<-matrix(rnorm(n*k),k)
x12<-apply(x12,1,mean)
hist(x12); mean(x12); var(x12)

## [1] -0.001660489
## [1] 0.03322667
  1. n=100
k<-1000
n<-100
x13<-matrix(rnorm(n*k),k)
x13<-apply(x13,1,mean)
hist(x13); mean(x13); var(x13)

## [1] -0.00240579
## [1] 0.0101806

Eksponensial

  1. n=10
k<-1000
n<-10
x21<-matrix(rexp(n*k),k)
x21<-apply(x21,1,mean)
hist(x21); mean(x21); var(x21)

## [1] 0.9997101
## [1] 0.10675
  1. n=30
k<-1000
n<-30
x22<-matrix(rexp(n*k),k)
x22<-apply(x22,1,mean)
hist(x22) ; mean(x22); var(x22)

## [1] 1.00212
## [1] 0.03377353
  1. n=100
k<-1000
n<-100
x23<-matrix(rexp(n*k),k)
x23<-apply(x23,1,mean)
hist(x23); mean(x23); var(x23)

## [1] 1.002253
## [1] 0.009846126

Seragam

  1. n=10
k<-1000
n<-10
x31<-matrix(runif(n*k),k)
x31<-apply(x31,1,mean)
hist(x31); mean(x31); var(x31)

## [1] 0.5021607
## [1] 0.007851549
  1. n=30
k<-1000
n<-30
x32<-matrix(runif(n*k),k)
x32<-apply(x32,1,mean)
hist(x32); mean(x32); var(x32)

## [1] 0.5006056
## [1] 0.002620425
  1. n=100
k<-1000
n<-100
x33<-matrix(runif(n*k),k)
x33<-apply(x33,1,mean)
hist(x33); mean(x33); var(x33)

## [1] 0.4999946
## [1] 0.0008358508
par(mfrow=c(3,3))
hist(x11);hist(x12);hist(x13);
hist(x21);hist(x22);hist(x23);
hist(x31);hist(x32);hist(x33);

Mean Var Mean Var Mean Var
n=10 n=10 n=30 n=30 n=100 n=100
Normal -0.01131697 0.2084703 0.004817455 0.1031404 0.00220261 0.06502746
Eksponensial 0.994701 0.09793023 1.000782 0.03524243 1.006847 0.01020049
Seragam 0.5029538 0.007959916 0.4989947 0.002710277 0.4984791 0.0008543492

Populasi Terhingga

y1<-rnorm(100000000)
y2<-rexp(100000000)
y3<-runif(100000000)
hist(y1); mean(y1); var(y1)

## [1] 0.0001116019
## [1] 1.000124
hist(y2); mean(y2); var(y2)

## [1] 1.000011
## [1] 1.000006
hist(y3); mean(y3); var(y3)

## [1] 0.5000063
## [1] 0.0833317
  1. n=10
k<-1000
n<-10
z11<-matrix(sample(y1, n*k),k)
z21<-matrix(sample(y2, n*k),k)
z31<-matrix(sample(y3, n*k),k)
z11<-apply(z11,1,mean)
z21<-apply(z21,1,mean)
z31<-apply(z31,1,mean)
hist(z11)

mean(z11)
## [1] 0.01084524
var(z11)
## [1] 0.09403203
hist(z21)

mean(z21)
## [1] 0.9944872
var(z21)
## [1] 0.09967982
hist(z31)

mean(z31)
## [1] 0.498567
var(z31)
## [1] 0.008284279
  1. n=30
k<-1000
n<-30
z12<-matrix(sample(y1, n*k),k)
z22<-matrix(sample(y2, n*k),k)
z32<-matrix(sample(y3, n*k),k)
z12<-apply(z12,1,mean)
z22<-apply(z22,1,mean)
z32<-apply(z32,1,mean)
hist(z12)

mean(z12)
## [1] 0.004068884
var(z12)
## [1] 0.03437383
hist(z22)

mean(z22)
## [1] 0.9962407
var(z22)
## [1] 0.03510297
hist(z32)

mean(z32)
## [1] 0.4991689
var(z32)
## [1] 0.002631119
  1. n=6
k<-1000
n<-100
z13<-matrix(sample(y1, n*k),k)
z23<-matrix(sample(y2, n*k),k)
z33<-matrix(sample(y3, n*k),k)
z13<-apply(z13,1,mean)
z23<-apply(z23,1,mean)
z33<-apply(z33,1,mean)
hist(z13)

mean(z13)
## [1] 0.00128341
var(z13)
## [1] 0.009966435
hist(z23)

mean(z23)
## [1] 0.9992907
var(z23)
## [1] 0.009980119
hist(z33)

mean(z33)
## [1] 0.5007172
var(z33)
## [1] 0.0008380591
par(mfrow=c(3,3))
hist(z11);hist(z12);hist(z13);
hist(z21);hist(z22);hist(z23);
hist(z31);hist(z32);hist(z33);

Mean Var Mean Var Mean Var
n=10 n=10 n=30 n=30 n=100 n=100
Normal -0.003441893 0.1039515 0.004621302 0.03133142 -0.003152491 0.009681597
Eksponensial 0.9843099 0.09396665 0.9905376 0.03701491 0.9998536 0.009838967
Seragam 0.4971369 0.008374428 0.4986376 0.002845931 0.5004413 0.0008938182