If Y and X are ind. binomial RV with parameters (n, p) and (a, b) respectively, then (Y/n) - (X/a) is approximately distributed. Find V(Y/n - X/a).

Question

I tried to find E(Y/n - X/a) and said it was E(Y/n) - E(X/a)= p - b. But then I got stuck finding the variance, I wasn't sure if it needed to be done with moment generating functions or if the Central Limit Theorem should be applied?

Answer 1

Because $X$ and $Y$ are independent, so are $X/a$ and $Y/n$. Thus, $$ V(Y/n-X/a)=V(Y/n)+V(X/a)=\frac{1}{n^2}V(Y)+\frac{1}{a^2}V(X)\\ =\frac{np(1-p)}{n^2}+\frac{ab(1-b)}{a^2}=\frac{p(1-p)}{n}+\frac{b(1-b)}{a}. $$