= Solution
For a <Poisson distribution> of parameter $\lambda$, its <probability generating function> is $\exp\{\lambda(z-1)\}$. Applying the <law of total expectation> to the <Poisson mixture> gives
$$
G_N(z)=\mathbb E\{\mathbb E(z^N\mid\lambda)\}=\mathbb E e^{(z-1)\lambda}.
$$
Thus
$$
\boxed{G_N(z)=M_\lambda(z-1).}
$$
For $|z|\le1$ the absolute value of the integrand is bounded by one, so this identity always exists as a <Laplace transform> of the positive mixing variable. An extension to $z>1$ requires the corresponding <moment-generating function> to be finite; the conditional computation itself does not guarantee positive exponential moments.
Back to article page
