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

Given Λ, X is Poisson with mean Λ. Λ has mean 3 and variance 2. What is Var(X)?

Var(X) is 5. By the law of total variance, the expected conditional variance is E[Λ] = 3 for a Poisson, and the variance of the conditional mean is Var(Λ) = 2. Adding these gives 5, which exceeds the mean, so X is overdispersed.

  1. A2
  2. B3
  3. C5Correct
  4. D6
  5. 9

Explanation

Var(X) = E[Var(X|Λ)] + Var(E[X|Λ]) = E[Λ] + Var(Λ) = 3 + 2 = 5. Answering 3 ignores the variation in the mean, which is the second term.

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