Question

Problem 1.29. Prove the central limit theorem for a sequence of i.i.d. Bernoulli(p) random variables, where p e (0,1). Hint: Compute the moment generating function of the object you want the limit of and use Taylor's expansion to show that it converges to the moment generating function of a standard normal. (In fact, the same proof, but without the computation being so explicit, works for a general distribution, as long as the secono moment is finite. And then pushing the proof a bit further leads to the Lindeberg-Feller CLT.)

Answer 1

Here we assume that we have a sample of size n from a binomial distribution for which probability of success of each r.v is p (as we assume i.i.d r.v's) then the sum of n r.v's follow binomial distribution. now for large n, we will prove CLT as,

therefore for large n mean binomial tends to normal distribution.