I have a probability question. I am struggling with a problem about finding the expectation of a compositions of PDF's. I know that expectation of a PDF is $$E(X)=\sum x p(x, \lambda)$$ I am not sure how to handle if the parameter itself is a PDF too. Here's the question: Give Poisson distribution $p(x, \lambda)=e^{-\lambda} \frac{\lambda^x}{x!}$ with parameter $\lambda$... but $\lambda$ is also a random variable with PDF of $f(\lambda) = e^{-\lambda}$... Find the Expectation.
I have a couple of ideas: 1) just make the "mother pdf" written in 2 variables. 2) somehow find expectation of Y, then plug that into "mother pdf" and find its expectation. I've tried both, but don't seem to be getting anywhere.
Any suggestions would be helpful.
Given $\lambda=\ell$, if $ e^{-\ell} \frac{\ell^x}{x!}$ can be considered as the pmf of the corresponding conditional distribution, then the conditional expectation is $\ell$. So, the expectation is $$\int_0^{\infty}\ell e^{-\ell}\ d\ell=1.$$