Computing the conditional expectation of $E[t^Y|X=k]$?

Question

For $X\in $Bin($n,p$) and $Y|X=k \in$Bin($k,p$), I have the conditional expectation, $$ E[t^Y|X=k]=\dots $$ I know that I probably should use the formula; $$ E[Y|X]=\sum_{y}y \cdot f(y|x), $$ But how do I compute the conditional expectation? I never get the correct answer $(1-p+pt)^k$.

Answer 1

In general, $$E[g(X)] = \sum_x g(x)P(X = x)$$ in the discrete case.

I will change the notation a little. In your case, if $X\sim \text{Bin}(m,p)$, and $Y|X\sim\text{Bin}(X,p)$, then \begin{align*} E[t^Y|X = n] = \sum_{k=0}^n t^k P(Y = k|X = n) = \sum_{k = 0}^n t^k\cdot\binom{n}{k}p^k(1-p)^{n-k}. \end{align*} In other words, don't use $$E[Y|X]=\sum_{y}y \cdot f(y|x).$$ Try to take it from here.