Comparisons of the generators

Question

Response to the following problem:

An antiquated generator for the normal distribution is:

Generate  U₁,.....,U₁₂ ˜u[-1/2 ,1/2]
Set Z  = Σ ¹² _i=1   Ui

the argument being that the CLT normality is sufficiently accurate with 12 terms.

a. Show that E[Z]=0 and var(Z)=1.

b. Using histograms, compare this CLT-normal generator with the Box-Muller algorithm. Pay particular attention to tail probabilities.

c. Compare both of the generators in part a. with rnorm.