 Solution

ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2014/3/b/ii/solution

The mean of a shape-rate gamma distribution is shape divided by rate. Hence the posterior mean is

E (θ ∣ x) = \frac{a + n x}{b + n} = \frac{b}{b + n} \frac{a}{b} + \frac{n}{b + n} \overline{x} .

Both weights are positive and sum to one. This is a convex combination of the prior mean and the sample mean, proving

min {a / b, \overline{x}} \leq E (θ ∣ x) \leq max {a / b, \overline{x}} .

For

n \geq 1

it lies strictly between them if the two means differ, and equals both if they agree. The rate

b

acts as the prior weight relative to the

n

observed exposure units.

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