IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Compound distributions and their applications in risk modelling
An insurer models aggregate claims S = X1 + ... + XN, where N is the number of claims and the Xi are independent, identically distributed claim amounts, independent of N. Which statement about this compound model is correct?
The mean of a compound random sum is the expected number of claims multiplied by the expected claim size, E[S] = E[N]E[X]. This follows from conditioning on N, since given N = n the expected total is n times the mean claim.
- AE[S] = E[N] × E[X]Correct
- BE[S] = E[N] + E[X]
- CE[S] = Var(N) × E[X]
- DE[S] = E[N] × E[X²]
- E[S] = E[X] / E[N]
Explanation
Conditioning on N gives E[S|N] = N E[X], so E[S] = E[N]E[X]. Adding the means or using E[X²] confuses the structure of a random sum.
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