Conditional Expectation of Poisson variables

Question

Let $X \sim exp(\lambda)$ be a random variable. For a given value of $X$, let $Y,Z \sim \operatorname{Poisson}(X)$ be two Poisson random variables. I try to calculate $E(Y\mid Z)$. I tried first to find $E(Y\mid X),E(X\mid Z)$. The first term is trivial, but I'm not sure how to find out $E(X\mid Z)$ (actually I'm not even sure that these calculations give the right answer at the end. Does total expectation law simply solve it?)

Answer 1

Let $d\mu(t)=\lambda e^{-\lambda t}dt$ $$P(Z=n|X)=e^{-X}\frac {X^{n}} {n!}.$$ So $$P(Z=n, X \leq x)= {\int_0^{x} {e^{-t} t^{n}}/n!} d\mu(t).$$ This gives $$P( X\leq x|Z=n)=\frac {{\int_0^{x} {e^{-t} t^{n}}/n!} d\mu(t)t} {{\int_0^{\infty} {e^{-t} t^{n}}/n!} d\mu(t)}.$$ Finally $$E(X|Z=n)=\frac {{\int_0^{\infty } t{e^{-t} t^{n}}/n!} d\mu(t)} {{\int_0^{\infty} {e^{-t} t^{n}}/n!} d\mu(t)}.$$

$E(Y|Z)$ cannot be calculated with the given information. However $E(Y|Z)=EY=EX=\lambda$ if you assume that $Y$ and $Z$ are independent.