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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 has a Poisson distribution with mean Λ, where Λ is uniformly distributed on (0, 4). What is Var(N)?

Var(N) is 10/3. By the conditional variance formula, Var(N) = E[Λ] + Var(Λ) for a Poisson with random mean. Here E[Λ] = 2 and Var(Λ) = 4/3, so the total is 2 + 4/3 = 10/3.

  1. A2
  2. B4/3
  3. C10/3Correct
  4. D3
  5. 14/3

Explanation

E[Λ] = 2 and Var(Λ) = 16/12 = 4/3. Var(N) = E[Var(N|Λ)] + Var(E[N|Λ]) = E[Λ] + Var(Λ) = 2 + 4/3 = 10/3. Option 4/3 ignores the process variance part; option 2 ignores the variance of Λ.

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