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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-36/1/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 1 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

With a uniform beta distribution prior, the posterior predictive distribution for another batch of the same size is

p (y ∣ x) = (y m) \frac{B ( x + y + 1 , 2 m - x - y + 1 )}{B ( x + 1 , m - x + 1 )} .

(1)

But

B (x + 1, m - x + 1) = 1/ [(m + 1) (x m)]

. Hence

p (y ∣ x) = (m + 1) (x m) (y m) B (x + y + 1, 2 m - x - y + 1) = p (x ∣ y) .

(2)

The displayed expression is symmetric in

x, y

. This is the uniform-count predictive property: the joint batch counts are exchangeable and both marginal counts are uniform. Conditional reversal is not a general property of arbitrary priors; it holds here because those marginal probabilities coincide.

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