Suppose $X|K = w$ is a Negative Binomial with parameters $r$ and $q$. K follows a Binomial Distribution with parameters $m$ and $p$.
I want to calculate the expected value of $$Z = max(X_1, X_2,...,X_n)$$.
I know the conventional way of solving such a problem using CDF of X. However, due to this rare condition, the CDF of X comes out to be very complex. I tried using the PGF of X too, but that also doesn't have a decent expression. Is there any simple algebra trick that could help me in solving such expression?