Expected Value functions of two randon variables

Question

$X$ and $Y$ are two independent random variables. $f$, $g$ and $h$ are 3 functions.

Can the below expected value be calculated?

$$E\left[ f(X)\sum_{k=0}^{\lfloor g(X) \rfloor }h(Y) \right]$$

$\lfloor g(X) \rfloor$ is capped by $n$. So I was thinking I could calculate the $n$ cases separately, and then average the results (weighted by their probability).

Thanks for any help.

Answer 1

Let $Z = f(X) \sum_{k=0}^{\lfloor g(X) \rfloor} h(Y)$. Then $E[Z|X] = f(X) (1+\lfloor g(X) \rfloor) E[h(Y)]$, so $$E[Z] = E \left[ f(X) (1+\lfloor g(X) \rfloor) \right] E[h(Y)]$$