Skip to content

IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Compound distributions and their applications in risk modelling

For a compound random variable S = X1 + ... + XN, where N is the claim number and the Xi are independent and identically distributed claim amounts independent of N, which expression gives E[S]?

The mean of a compound random variable is the expected number of claims multiplied by the expected claim size, E[S] = E[N]E[X]. This follows by conditioning on N, since each of the N claims contributes mean E[X] given independence.

  1. AE[N] × E[X]Correct
  2. BE[N] + E[X]
  3. CVar(N) × E[X]
  4. DE[N] × E[X²]
  5. E[N²] × E[X]

Explanation

Conditioning on N gives E[S|N] = N·E[X], so E[S] = E[N]E[X]. Adding the means is wrong because S is a random sum of N terms, not a sum of two quantities. Using E[X²] would give a second-moment quantity instead of a mean.

Did you get it right without looking?

One question tells you little. A timed set on Compound distributions and their applications in risk modelling shows your real accuracy, how long you take and where you lose marks.

Practise Compound distributions and their applications in risk modellingFree diagnostic mock

More Compound distributions and their applications in risk modelling questions