Question

Question 1, 2

or similar to demonstrate the central limit theorem. sample mean when the population distribution is expo- pling distribution changes as n, the sample size increases. In this homework assignment you will use R (or python or similar) to demo . First we will explore the sampling distribution of a sample mean when the nential (X, 1 exp(1)). We will show how the sampling distribution changes a - X, id exp(1) and n=1 * In a document (word or latex or similar) you will answer the numbered the theoretical mean and standard deviation of X ? * Take 10,000 samples with a sample size of n=1. * (2) what are the mean and standard deviation of X, as cale and standard deviation of X, as calculated from your 10,000 samples? moothed density plot Inold . imilar) you will answer the numbered questions. (1) what are

Answer 1

if $X \sim exp (1)$ Then the mean E(X)=1 and standard devaition is also 1.

simulation of exp(1) is carried out using R

The R code and output is as follows

n=10000
set.seed(1)
for(i in 1 :n)
{
x[i]=rexp(1,1)
mean[i]=mean(x)
sd[i]=sd(x)
}
mean
sd
##########################################################

From the output given above we can see that the mean and standard deviation are close to the theoritical value.

Give thumbs up if you are satisfied else please comment for further clarifications