Let X∼Poisson(λ)X \sim \text{Poisson}(\lambda)X∼Poisson(λ). What is the characteristic function ϕX(t)=E[eitX]\phi_X(t) = E[e^{itX}]ϕX(t)=E[eitX]?
eλ(eit−1)e^{\lambda(e^{it}-1)}eλ(eit−1)
e−λt2/2e^{-\lambda t^2/2}e−λt2/2
(1−it/λ)−λ(1 - it/\lambda)^{-\lambda}(1−it/λ)−λ
exp(itλ−λt2/2)\exp(it\lambda - \lambda t^2/2)exp(itλ−λt2/2)