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Past exam of the mathematics course of the University of Cambridge

2024

Paper 4

Campbell's formula for a <Poisson point process> gives
$$
\mathbb ES=\int_{\mathbb R^d}g(u)\lambda(u)du.
$$
Expanding $S^2$ separates equal and distinct Poisson points. The factorial-moment formula gives
$$
\mathbb ES^2
=\int g(u)^2\lambda(u)du
+\left(\int g(u)\lambda(u)du\right)^2.
$$
Subtracting the square of the mean yields
$$
\boxed{\operatorname{var}S=\int_{\mathbb R^d}g(u)^2\lambda(u)du.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 27K / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 27K / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 27K / c / Solution

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