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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 2

The sample mean $S_n/n$ has expectation $p$ and variance
$$
\frac{p(1-p)}n\leq\frac1{4n}.
$$
The sum in the question is exactly
$$
\mathbb P\left(\left|\frac{S_n}{n}-p\right|>\delta\right).
$$
Chebyshev's inequality therefore gives, uniformly in $p$,
$$
\boxed{
\sum_{k\in K_\delta}\binom nkp^k(1-p)^{n-k}
\leq\frac1{4n\delta^2}}.
$$


Solution

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

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

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

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