= Covariance of two components of a compound Poisson sum
{title2=$\operatorname{Cov}(\sum_{i=1}^NY_i,\sum_{i=1}^NZ_i)=\lambda\mathbb E[Y_1Z_1]$}
Let $N\sim\operatorname{Pois}(\lambda)$ be <independent> of <independent and identically distributed random variables> $(Y_i,Z_i)$ with finite second <moments>. Conditioning on $N$ gives <covariance> $N\operatorname{Cov}(Y_1,Z_1)$ and <conditional expectations> $N\mathbb EY_1,N\mathbb EZ_1$. <Law of total covariance> and $\operatorname{Var}N=\lambda$ yield the formula. For per-claim <excess of loss reinsurance> payouts $Y=\min(X,M)$ and $Z=(X-M)_+$, it becomes $\lambda M\mathbb E[(X-M)_+]$. The insurer and reinsurer aggregates share their claim count and are generally dependent.
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