Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 31 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 28K / c / Solution

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{wiki=Campbell's_theorem_(probability)}

For a Poisson point process $\Pi$ with intensity measure $\mu$ and a square-integrable measurable function $g$,
$$
\mathbb E\sum_{x\in\Pi}g(x)=\int g\,d\mu,
\qquad
\operatorname{Var}\left(\sum_{x\in\Pi}g(x)\right)=\int g^2\,d\mu.
$$
The second identity follows because distinct points contribute the square of the mean, while the diagonal contribution is $\int g^2\,d\mu$.


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