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

Let N be Poisson with mean 3, and given N = n, S is the sum of n independent claims each with mean 100 and variance 400 (S = 0 if n = 0). What is Var(S)?

Var(S) is 31,200. By the law of total variance, it is 3×400 for claim variability plus 3×100² for variation in claim number, giving 1,200 + 30,000.

  1. A30,000
  2. B31,200Correct
  3. C40,000
  4. D1,200
  5. 9,000

Explanation

Var(S) = E[Var(S|N)] + Var(E[S|N]) = E[N]×400 + Var(N)×100² = 3×400 + 3×10,000 = 1,200 + 30,000 = 31,200. Option 30,000 omits the within-N variance term; 1,200 omits the variation in N.

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