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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-35/2/e/3/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 2 / e / 3 / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

The next Beta distribution has conditional expected value and variance

E [p_{T} ∣ X] = \frac{α + X}{s + n}, Var (p_{T} ∣ X) = \frac{( α + X ) ( β + n - X )}{( s + n ) ^{2} ( s + n + 1 )} .

(1)

For large

n

, its expected value approaches

X / n

, while its variance is at most

1/ [4 (s + n + 1)]

. The final draw therefore stays close to the first. These are prior distributions for the cup probabilities: the auxiliary count is a device for constructing dependence, rather than additional observed tea data.

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