Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 28L / a

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- Solution a

Solution

For an observed value

X = x

, the likelihood is

p (x ∣ θ) = \frac{e ^{- θ} θ ^{x}}{x !},

while the gamma prior with shape

α

and rate

λ

π (θ) = \frac{λ ^{α}}{Γ ( α )} θ^{α - 1} e^{- λ θ}, θ > 0.

The posterior density is therefore proportional to

p (x ∣ θ) π (θ) \propto θ^{α + x - 1} e^{- (λ + 1) θ} .

Normalizing this gamma kernel proves the Poisson-gamma conjugacy formula

θ ∣ X = x \sim Gamma (α + x, λ + 1) .

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