Q:

Quick question about Binomial variance

I am having trouble figuring a problem if we have T ~ uniform on {1,2,3,...,10} and X ~ Bin(T,0.5) and we have to find the variance of X, would that be mean*.5*.5(I found the mean to be 2.75)?

A:

Hint: To do the calculation, it is useful to know $E(X^2)$. We have
 $$E(X^2\mid T=k)=\frac{1}{4}(k+k^2).$$
It follows that
$$E(X^2)=\sum_{k=1}^{10} \frac{1}{10}\cdot\frac{1}{4}\cdot(k+k^2).$$

