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

Let

S_{g} = \sum_{x \in Π} g (x) .

The compact support of

g

ensures that this sum has only finitely many nonzero terms almost surely. The intensity measure of the ideal gas is

μ (d x) = z d x .

Applying the first-moment identity in Campbell theorem gives

E S_{g} = z \int_{R^{3}} g (x) d x .

For the second moment, separate equal and distinct particles:

E S_{g}^{2} = E x \in Π \sum g (x)^{2} + E x, y \in Π x \neq = y \sum g (x) g (y) = z \int_{R^{3}} g (x)^{2} d x + z^{2} (\int_{R^{3}} g (x) d x)^{2} .

Subtracting

(E S_{g})^{2}

yields

Var (S_{g}) = z \int_{R^{3}} g (x)^{2} d x .

These are the mean and variance formulas in Campbell's theorem for a homogeneous Poisson point process.

Solved by gpt-5.6-sol high.

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