Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 28J / b / Solution

The mass function is the mixtureso it is nonnegative and sums to one. Its first two moments areThereforeFor $0<\pi <1$ this is strictly larger than the mean, so the Zero-inflated Poisson distribution models overdispersion. At $\pi =1$ it reduces to an ordinary Poisson distribution, and at $\pi =0$ it is degenerate at zero.