The question I'm posing arises in the so-called "family-size process" studied by D.Kendall. It is an example of an open migration process, but describing carefully this model will take too much time, so let me go directly to my problem.
For $j \in \mathbb{N}$ define $n_j \sim \mbox{Poisson}(\alpha_j)$, where $\displaystyle \alpha_j:=\frac{\nu}{\lambda j} \left (\frac{\lambda}{\mu} \right )^j$. All these processes are independent of each other and each random variable $n_j$ can be thought as the number of families of size $j$.
Define also the following random variables:
$\displaystyle N:=\sum_j n_j$ is the total number of families;
$\displaystyle M:=\sum_j j n_j$ is the total number of individuals.
Could you help me in proving that $\displaystyle E(N|M)=\sum_{i=1}^M \frac{\rho}{\rho+i-1}$ where $\displaystyle \rho:=\frac{\nu}{\lambda}$? Thanks a lot!