= Poisson size-bias identity
{c}
{title2=$\mathbb E[N h(N-1)]=\lambda\mathbb E[h(N)]$}
For $N$ with <Poisson distribution> of mean $\lambda$, the equality $n\mathbb P(N=n)=\lambda\mathbb P(N=n-1)$ proves the identity by summation. For an independent <random sum of independent claims> $S$ with this count, exchangeability of the summands gives $\mathbb E[S h(S)]=\lambda\mathbb E[X h(S+X)]$, where $X$ on the right is independent of $S$ and has the severity law. These identities hold for nonnegative measurable functions, and for integrable signed functions. They obtain <moments> without assuming that a positive <exponential moment> exists.
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