Question

2) Illustrating the central limit theorem. X, X, X, a sequence of independent random variables with the same distribution as X. Define the sample mean X by X = A + A 2 be a random variable having the exponential distribution with A -2. Denote by -..- The central limit theorem applied to this particular case implices that the probability distribution of converges to the standard normal distribution for certain values of u and o (a) For what values of μ and σ does the convergence hold? (b) For each of the four values n = 5, 10, 20 and 50 do the following: Obtain 10,000 independent values of the sample mean random variable X, where the X, are generated from the exponential distribution with A 2; then create a histogram of for each n. Discuss what you notice for each case on n

R commands

Answer 1

a. $\mu=0.5 \ and\ \sigma=0.5$ ,

b.

w-c () for # n-) 2 for (i in 1:10000) W-c(x,mean(rexp(5,2))) 4 hist (w) # 5 x-o() for n-10 7 for (i in 1:10000) 8 X-c(x,mean (rexp (10,2))) 9 10 hist (x) 11 y-c() # for n-20 12 for (i in 1:10000) 13 y-x,mean(rexp (20,2))) 14 15 hist(x) # for z-c() n-50 16 17 for (i in 1:10000) 18 z-c(x, mean(rexp (50,2))) 19 20 hist (z)