Question

R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which we can see should be 100, For now, let the sample size be n = 3, Pick 3 random numbers from X, compute X., and repeat the process a total of 50000 times. Plot a smooth version of the histogram of these 50000 values for X: the plot(density(.. command in R wll be useful. Now find the average of your 50000 values and make a vertical dotted line in R at this number (match the color of your curve). You have just made a rough picture of the density function for (when n 3) and identified its (approximate) expected value Now you should repeat this process for n 6, 10 andn 50. Plot all four of these curves (use different colors) and vertical lines solid vertical line at 100. You should be able to see the asymptotic unbiasedness in the picture. Your answer is your code and a rough sketch of the plot you have created

Answer 1

The following is the R code (all statements starting with # are comments)

get this plot

We can see that as the sample size increases, the dotted vertical line moves closer to the solid line corresponding to $\mu^2=100$, showing the asymptotic unbiasedness

We can also observe that the variance of the sampling distribution reduces with the increase in sample size.

Finally we can also see that the density takes on a bell shape as the sample size increases, indicating the asymptotic normality.

The code in text format is below

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#set the mean
mu<-10
# set the standard deviation
sigma<-8
#set the random seed
set.seed(123)
#make an empty plot to add the curves later
plot(1, type="n", xlab=bquote(bar(X)^2), ylab="Density", xlim=c(0, 800),ylim=c(0,0.02),
   main=bquote("The distribution of "*bar(X)^2))
#set the colors for the plot
cols<-c("red","blue","green","magenta")
#set the sample sizes
n<-c(3,6,10,50)
#do this for each sample size n
for (i in 1:length(n)){
   #set the number of samples to be taken
   r<-50000
   #draw a sample of size n, r times, that is n*r draws from normal(10,8^2)
   x<-rnorm(n[i]*r,mu,sigma)
   #make this into a matrix of n rows and r columns
   x<-matrix(x,nrow=n[i],ncol=r)
   #get r sample means
   xbar<-apply(x,2,mean)
   #get the square of means
   xbar2<-xbar^2
   #draw the density of xbar2
   lines(density(xbar2),typ="l",col=cols[i])
   #add the vertical line
   abline(v=mean(xbar2),col=cols[i],lty=2)
}
#add a solid line for mu^2
abline(v=mu^2,lwd=2)
#add the legend
legend("topright",paste("n=",n,sep=""),col=cols,lty=1)