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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 2 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

The difference of the Bernoulli distribution variances factors as

v_{0} - v_{1} = p_{0} (1 - p_{0}) - p_{1} (1 - p_{1}) = (p_{0} - p_{1}) (1 - p_{0} - p_{1}) .

(1)

The first factor is positive, whereas

p_{0} > 1 - p_{1}

makes the second negative. Thus

v_{0} < v_{1}

and

R^{*} = v_{0} / v_{1} < 1

. Therefore Neyman allocation assigns more patients to treatment 1, the inferior treatment. Minimizing the variance of a treatment comparison and maximizing the number of desirable responses are different objectives.

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