One of the earlier methods (not one of the better ones) of generating pseudo-random standard normal random variables from uniform random variables is to take X = ∑12i= l Ui - 6, where the U,s are iid uniform(0,1).
(a) Justify the fact that X is approximately n(0,1).
(b) Can you think of any obvious way in which the approximation fails?
(c) Show how good (or bad) the approximation is by comparing the first four moments. (The fourth moment is 29/10 and is a lengthy calculation-mgfs and computer algebra would help; see Example A.0.6 in Appendix A.)