USE R TO SOLVE

Consider a random sample X1, ....Xn from an Exponential distribution with mean

and define by Y

(a) State the distribution of Y , giving all the parameters of the distribution.

(b) Fix the seed value using set.seed(2021). Generate a value of from the Uniform U(0.1,2.0)

, rounded to one decimal place. Use this value in parts (c) to (g).

(c) Fix the seed value using set.seed(2021). Perform simulation of a sample X1...Xn

with sample size n=15 from the Exponential distribution with mean .

(d) Calculate the value of Y for the sample in part (c).

(e) Fix the seed value using set.seed(2021) at the beginning. Then, perform 1,000

repetitions of parts (c) and (d) to obtain sample y1...yB from the random

variable Y with B=1000 .

(f) Plot a histogram showing the relative frequencies of the sample from part (e).

(g) (i) Compare graphically the histogram in part (f) to the density of a suitable

Normal distribution.

(ii) Comment on your findings in part (g)(i) in the context of the Central Limit Theorem