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

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- Solution b

Solution

The mass function is the mixture

p_{π, λ} = (1 - π) δ_{0} + π Poisson (λ),

so it is nonnegative and sums to one. Its first two moments are

E [Y] = πλ, E [Y^{2}] = π (λ + λ^{2}) .

Therefore

Var (Y) = πλ + π (1 - π) λ^{2} .

For

0 < π < 1

this is strictly larger than the mean, so the Zero-inflated Poisson distribution models overdispersion. At

π = 1

it reduces to an ordinary Poisson distribution, and at

π = 0

it is degenerate at zero.

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