***Please answer in R code / R studio***

Problem 3 (40 points) Assume X1, X,,.., X, iid follow N (1,0%). (a) State the distribution of the r.v. Yn = 2Xf and give the value of its parameters. 1 (b) Plot the probability density functions of randomn variables X, and different Yo on the same plot. Use p=2, o=1. Plot the Yn density for n = 5, 10,30,50. (c) By Central Limit Theorem, construct a new r.v. Z that approximates N(0, 1) using X, X, X for n +0. To do this please write code that: (a) Generate X1, X3...., X.. (b) Calculate z for that n value. Do this sequentially to obtain a sequence of Z values. Then plot this sequence where the ordinate is n and the corresponding abscissas are 2 values. Finally add a horizontal line for the value of je on the same plot. Problem 3 (40 points) Assume X1, X,,.., X, iid follow N (1,0%). (a) State the distribution of the r.v. Yn = 2Xf and give the value of its parameters. 1 (b) Plot the probability density functions of randomn variables X, and different Yo on the same plot. Use p=2, o=1. Plot the Yn density for n = 5, 10,30,50. (c) By Central Limit Theorem, construct a new r.v. Z that approximates N(0, 1) using X, X, X for n +0. To do this please write code that: (a) Generate X1, X3...., X.. (b) Calculate z for that n value. Do this sequentially to obtain a sequence of Z values. Then plot this sequence where the ordinate is n and the corresponding abscissas are 2 values. Finally add a horizontal line for the value of je on the same plot