Let 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)
es(λ−1)e^{s(\lambda-1)}es(λ−1)
sλs^\lambdasλ
11−λs\frac{1}{1-\lambda s}1−λs1