Binomial to standard normal using the central limit theorem?

Question

I need to prove that:

[X_n - E(X_n)]/[var(X_n)

tends in distribution to a standard normal distribution, where

X_n~Bin(n,p)

I know that E(X_n) = np, var(X_n) = np(1-p), and letting Y_n equal what i need to show, and finding the mgf of Y_n, i get

[p(exp{t/sqrt(np(1-p))) + (1-p)]^n times exp{-npt/sqrt(np(1-p))}

Where sqrt is the square root. I have taken logs of both sides, to get

[-n^2pt/sqrt(np(1-p))] log [p(exp{t/sqrt(np(1-p))) + (1-p)]

Now i think i need to use the taylor expansions for exp(x) then for log(1+x), but this doesn't simplify to exp(t^2/2) which is what i need to show this expression does simplify to. Am i on the right tracks, or am i totally wrong? If i'm wrong, could you point me in the right direction, step by step please? Thanks

Answer 1

It should be sqrt of var(X_n) in the denominator.

The question asks you to *use* the CLT, not to prove it. So all you need to do is note that your binomial random variable X_n is the sum of n independent Bernoulli (0-1 valued) vbls. This shows that the statement (convergence of X_n standardized to zero mean and unit variance) is a special case of the CLT.

Answer 2

My text shows that

(X-np)/sqrt(var) -> exp(t^2/2) as n->inf using the moment generating function.

And, exp(t^2/2) is also the value of

integral -inf to +inf of
{ (1/sqrt(2pi)*exp(tz)*exp(-z^2/2)dz }

It's not a trivial thing to show.