= Retained compound Poisson aggregate
{title2=$T_I=\sum_{j=1}^N g(X_j)$}
Applying a measurable per-claim retention $g$ to independent claim sizes preserves the <compound Poisson distribution> form. The transformed severity is the <pushforward measure> of the original severity law. With count parameter $\lambda$, the retained aggregate has <expected value> $\lambda\mathbb E g(X)$ and <variance> $\lambda\mathbb E[g(X)^2]$. Zero retained marks may be removed by <Poisson thinning>.
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