If X∼Poisson(λ)X \sim \text{Poisson}(\lambda)X∼Poisson(λ), what is the probability generating function G(s)=E[sX]G(s) = E[s^X]G(s)=E[sX]?
eλ(s−1)e^{\lambda(s-1)}eλ(s−1)
eλse^{\lambda s}eλs
11−λs\frac{1}{1-\lambda s}1−λs1
(1−s)−λ(1-s)^{-\lambda}(1−s)−λ