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

A portfolio has two equally likely risk classes. Class A: claim mean 10, variance 4. Class B: claim mean 16, variance 12. What is the unconditional variance of a claim chosen at random?

The variance is 17. The average within-class variance is 8, and the variance between the class means 10 and 16 (overall mean 13) is 9. The law of total variance adds these two components to give 17.

  1. A8
  2. B9
  3. C17Correct
  4. D14
  5. 12

Explanation

E[Var] = (4+12)/2 = 8. The conditional means are 10 and 16 with equal weights, so Var(E) = 0.5×(10-13)² + 0.5×(16-13)² = 9. The total is 8 + 9 = 17. Using only the average variance gives 8.

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