Stat 359 Assignment 2 Solutions
Thanks to Dr. Laura Cowen for preparing these solutions.
1.
Central Limit Theorem
# First initialize the size of the samples and the number of samples you want to take.
# Also initialize your matrices that will hold your sample means to zero.
size<-c(10,20,50,100)
samp<-c(10,100,1000)
u10<- matrix(rep(0,times=length(size)*10), nrow=length(size), ncol=10)
u100<- matrix(rep(0,times=length(size)*100), nrow=length(size), ncol=100)
u1000<- matrix(rep(0,times=length(size)*1000), nrow=length(size), ncol=1000)
b10<- matrix(rep(0,times=length(size)*10), nrow=length(size), ncol=10)
b100<- matrix(rep(0,times=length(size)*100), nrow=length(size), ncol=100)
b1000<- matrix(rep(0,times=length(size)*1000), nrow=length(size), ncol=1000)
p10<- matrix(rep(0,times=length(size)*10), nrow=length(size), ncol=10)
p100<- matrix(rep(0,times=length(size)*100), nrow=length(size), ncol=100)
p1000<- matrix(rep(0,times=length(size)*1000), nrow=length(size), ncol=1000)
# This part of the code could have be done various ways.
Here I take 3 loops that will
# produce 9 matrices of means.
For instance, u10 is a 4x10 matrix where the rows
# represent the size of the samples (10,20,50,100) and the columns are the 10 sampled
# means.
for (k in samp){
# k indicates the number of samples 10, 100,l000
index<-0
# counter for the rows in the matrices
for (i in size){
# i changes with size of the sample 10, 20, 50, 100
index<-index+1
# increase the row number by 1
for (j in 1:k){
# j indicates the sample number, 1 to sample size
if (k==10){
# when size of the sample is 10 to produce a 4x10 matrix
u10[index,j]<-mean(runif(i, min=0,max=1))
p10[index,j]<- mean(rpois(i, 5))
b10[index,j]<- mean(rbinom(i, 1, 0.20))
}
if (k==100){
# when size of the sample is 100 to produce a 4x100 matrix
u100[index, j]<- mean(runif(i, min=0,max=1))
p100[index, j]<- mean(rpois(i, 5))
b100[index, j]<- mean(rbinom(i, 1, 0.20))
}
if (k==1000){
# when size of the sample is 1000 to produce a 4x1000 matrix
u1000[index, j]<- mean(runif(i, min=0,max=1))
p1000[index, j]<- mean(rpois(i, 5))
b1000[index, j]<- mean(rbinom(i, 1, 0.20))
}
}
}
}
To conserve plots, put 12 on a page, 1 page for each distribution.
Make sure that for comparison
purposes the x axis are the same so that we can see how the distributions change.
Note that we
could also keep the y axis the same, however it may be too different from plot to plot.
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