Question

Conduct a computer simulation to generate the sample means of Poisson random variables. First generate X ∼ Poisson(3), and plot a histogram of this Poisson random variable. Then generate X = 1/5(X1 + X2 + X3 + X4 + X5), where X1, X2, . . . , X5 are all from Poisson(3). Plot a histogram of this sample mean statistic X. Compare the histograms and describe the changes you see in the histograms. Explain the changes using Central Limit Theorem. (For sample code, please refer to the course website. Please attach the codes and the two computer generated histograms. Use R version

Answer 1

ANSWER: I have attached the R Code for the given problem. We have two histograms, one is of the Poisson population with parameter 3. For that we have obtained 5000 realizations of Poisson(3) and plotted the histogram for it. The other histogram we have taken the mean of 5 observations from the sample and repeated this step 5000 times so that we get 5000 realizations of the sample mean. It is clear from the histogram of the sample mean that the sample mean distribution approaches normal distribution as n tends to infinity. Here n is small, i.e. 5. But if we take n to be greater than 30, the sample mean distribution will be normal, i.e.

$\bar{X} \sim N\left( \lambda, \frac{\lambda}{n}\right )$

as n>30.

####################### R CODE ##############################################

rm(list=ls(all=TRUE))

set.seed(12345)

n=5000
X=rpois(n,lambda = 3)
mean(X)
sd(X)

MX=rep()
for(i in 1:5000)
{
S=sample(X,5)
MX[i]=mean(S)
}

mean(MX)
sd(X)

par(mfrow=c(1,2))
hist(X)
hist(MX,main="Histogram of Sample Mean")

############################### END OF R CODE ############################################

Histogram of X Histogram of Sample Mean 髦3 41ゆ 12 12 12 髦3 12 し丈 0 2 4 6 8 10 MX