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
2006-12-10
09:15:12
·
2 answers
·
asked by
drummanmatthew
2
in
Mathematics