Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 9 / a / iv / Solution

The conditional mean of a Poisson distribution equals its parameter, so $\mathbb{E}[Z\mid X]=X$. The law of total expectation givesAs a check, differentiating the probability generating function in part (iii) at one also gives $G_Z^{\prime }(1)=\mu$.