Gamma-Poisson posterior mean for family size

The number of children per family is modelled as Poisson with rate $\lambda$, and $\lambda$ is given a Gamma prior with shape $32$ and rate $10$. A survey of $34$ families finds $100$ children in total. What is the posterior mean of $\lambda$?

$3$. The Gamma prior is conjugate to the Poisson: the posterior is Gamma with shape $32+100$ and rate $10+34$, so its mean is $\tfrac{132}{44}=3$.