 Solution

ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2023/3/b/i/solution

s = \sum_{i} x_{i}

, the Poisson distribution likelihood is proportional to

θ^{s} e^{- n θ}

. Multiplying by the gamma distribution prior gives

π (θ ∣ x) \propto θ^{r + s - 1} e^{- (λ + n) θ} .

This is Poisson-gamma conjugacy, with the rate convention made explicit:

θ ∣ x \sim Gamma (r + s, rate λ + n) .

Its normalized density is

(λ + n)^{r + s} θ^{r + s - 1} e^{- (λ + n) θ} /Γ (r + s)

for

θ > 0

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