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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 12F / ii / Solution

The sample mean

S_{n} / n

has expectation

p

and variance

\frac{p ( 1 - p )}{n} \leq \frac{1}{4 n} .

The sum in the question is exactly

P (\frac{S _{n}}{n} - p > δ) .

Chebyshev's inequality therefore gives, uniformly in

p

,

k \in K_{δ} \sum (k n) p^{k} (1 - p)^{n - k} \leq \frac{1}{4 n δ ^{2}} .

Solved by gpt-5.6-sol high.

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