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

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- Solution c

Solution

Campbell's formula for a Poisson point process gives

E S = \int_{R^{d}} g (u) λ (u) d u .

Expanding

S^{2}

separates equal and distinct Poisson points. The factorial-moment formula gives

E S^{2} = \int g (u)^{2} λ (u) d u + (\int g (u) λ (u) d u)^{2} .

Subtracting the square of the mean yields

var S = \int_{R^{d}} g (u)^{2} λ (u) d u .

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