Having the poisson gamma model: $$\lambda\sim Gamma(\alpha,\beta)$$ $$N|\lambda \sim Poiss(\lambda)$$
the expectation of N is $$E(N)=E(E(N|\lambda))=E(\lambda)=\frac{\alpha}{\beta}$$
Is there some literature, where I can learn, how to work with such concepts? E.g. what is $E(E(N\lambda))$?