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IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations

The number of claims N on a policy in a year is Poisson with mean 3. Each claim is independently of size with mean Rs 20,000 and variance Rs² 100,000,000. Total claims S is the sum of N claim sizes. What is Var(S) in Rs² (using the conditional variance formula)?

Var(S) is Rs² 1,500,000,000. The expected conditional variance is 3×100 million = 300 million, and the variance of the conditional mean is 20,000² ×3 = 1,200 million. Adding them gives 1,500 million.

  1. A300,000,000
  2. B1,200,000,000Correct
  3. C1,500,000,000
  4. D900,000,000
  5. 600,000,000

Explanation

E[Var(S|N)] = E[N]·σ² = 3×100,000,000 = 300,000,000. Var(E[S|N]) = Var(N·20,000) = 20,000²×3 = 400,000,000×3 = 1,200,000,000. Total = 1,500,000,000. So the key is 1,500,000,000, option index 2 only; the 1,200,000,000 option omits the first term.

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