Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 8 / a

The exponential series gives ${\sum }_{n=0}^{\infty }{\lambda }^n/n!=e^{\lambda }$. Thus the normalization constant for the Poisson distribution isEvery value $e^{-\lambda }{\lambda }^n/n!$ is nonnegative, and their sum is $e^{-\lambda }{e}^{\lambda }=1$. Hence this is a valid probability mass function.

By definition, the probability generating function isSubstituting the Poisson distribution and evaluating the exponential series givesThe series converges for every finite complex $s$; in particular it gives the probability generating function on $0\le s\le 1$.