= Campbell's theorem
{c}
{wiki=Campbell's_theorem_(probability)}
= Campbell theorem
{c}
{synonym}
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 $\int g^2\,d\mu$.
The expectation identity holds for a general <point process> with intensity measure $\mu$ whenever the sum is integrable. The stated variance identity uses the additional <Poisson point process> assumption.
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