Collective Risk Model

\[S_N=X_1+X_2+\cdots+X_N\]

Conditional Expectation and Variance

\[\begin{align*} E[X]&=E[E[X|Y]] \\ \mathrm{Var}(X)&=E[\mathrm{Var}(X|Y)]+\mathrm{Var}(E[X|Y]) \end{align*}\]

Compound Distribution

\[\begin{align*} E[S]&=E[N]E[X] \\ \mathrm{Var}(S)&=E[N]\mathrm{Var}(X)+\mathrm{Var}(N)E[X]^2 \end{align*}\]

Compound Poisson, \(\mathcal{CP}(\lambda,F_X)\)

If \(N\sim\mathcal{P}(\lambda)\), then \[\begin{align*} E[S]&=\lambda E[X] \\ \mathrm{Var}(S)&=\lambda E[X^2] \end{align*}\]