Question

Use r statistical software to

1. Pick a population with a particular distribution.

2. From the population use software to obtain k random samples (for example k = 10) each containing n elements (for example n = 30.)

3. Give the distributions of X̄

4. For each sample, calculate the value of the statistic and construct a histogram of the k values. This histogram gives the approximate sampling distribution of the statistic. The statics of interest are X̄ and V (X̄ )

5. Calculate E(X̄) and V (X̄)

Answer 1

We choose a exponential distribution with rate parameter . Thus . The distribution of the sample mean is

The histogram of sampling distribution is plotted below for n= 30, k= 100.

The histogram is approximately normal.

The empirical values of mean and variance are

The theoretical values are,

R code below:

set.seed(47757)
lambda <- 0.2
n <- 30
k <- 100
Xbar <- array(dim=k)
for (i in 1:k)
{
X <- rexp(n,lambda )
Xbar[i] <- mean(X)
}
plot(1:1)
dev.new()
hist(Xbar, freq=FALSE, main= "Histogram of sample means of Exponential distribution", lwd=2, col = "sky blue", xlab=expression(bar(X)), ylab = "Density")
mean(Xbar)
var(Xbar)