Conditional Expectation of joint/(composite function?)

Question

I'm prepping for an exam and looking through previous exams I came across this question:

Let $Z$ be a Poisson distributed stochastic variable with parameter $Λ$. In turn, $Λ$ is a Poisson distributed stochastic variable with parameter $\mu=2$. Compute $E[Z]$ (Hint: $Z|Λ=\lambda$~Po($\lambda$).)

answer: $E[Z]=2$

My first step was:

$E[Z]= E\Big[E[Z\;|\;Λ]\Big]$

however I'm not at all sure how to proceed. any ideas?

Answer 1

Next step is to evaluate $E[Z|\Lambda]$. By the hint, given that $\Lambda=\lambda$ we know that $Z$ has a Poisson($\lambda$) distribution and therefore $E(Z|\Lambda=\lambda)=\lambda$ (since the mean of a Poisson distribution equals its parameter). Conclude $E(Z|\Lambda)=\Lambda$.