 Heterogeneous individual claims model

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 $\mathbb{E}T={\sum }_i{q}_i{\mu }_i$ and $\mathrm{Var}(T)={\sum }_i[q_i{\sigma }_i^2+q_i(1-q_i){\mu }_i^2]$. 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.