 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/28j/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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

π = 1

it reduces to an ordinary Poisson distribution, and at

π = 0

it is degenerate at zero.

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