= Heterogeneous individual claims model
{title2=$T=\sum_i B_iY_i$}
Each policy has an independent <Bernoulli random variable> $B_i$ of probability $q_i$, and its independent severity $Y_i$ has <expected value> $\mu_i$ and <variance> $\sigma_i^2$. Independent policies give $\mathbb ET=\sum_iq_i\mu_i$ and $\operatorname{Var}(T)=\sum_i[q_i\sigma_i^2+q_i(1-q_i)\mu_i^2]$. Conditioning on the claim indicator proves the formulas. Keeping the policy label preserves the dependence between the occurrence probability and the severity law; replacing all labels by a single mixture is an approximation unless the relevant laws agree.
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