Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 2 / b / ii

For

L = n_{0} (1 - p_{0}) + n_{1} (1 - p_{1}) + η (\frac{a}{n _{0}} + \frac{b}{n _{1}} - C),

(1)

the stationary equations are

1 - p_{0} = \frac{η a}{n _{0}^{2}}, 1 - p_{1} = \frac{η b}{n _{1}^{2}} .

(2)

Dividing them gives

R^{*} = \frac{n _{0}}{n _{1}} = \frac{a ( 1 - p _{1} )}{b ( 1 - p _{0} )} = \frac{p _{0}}{p _{1}} \frac{1 - p _{1}}{1 - p _{0}} .

(3)

The fixed-power constraint then determines the total sample size. Strict convexity after eliminating one variable supplies the second-order minimum condition.

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 2 / b / ii

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