IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations
Given Λ, a risk has claim count N with Poisson distribution of mean Λ. Λ takes value 2 with probability 0.5 and value 6 with probability 0.5. What is Var(N)?
Var(N) is 8. By the law of total variance, the expected conditional variance E[Λ] = 4 plus the variance of the conditional mean Var(Λ) = 4 gives 8. Ignoring the variation in Λ would wrongly give 4.
- A4
- B6
- C8Correct
- D10
- 12
Explanation
E[Var(N|Λ)] = E[Λ] = 4. Var(E[N|Λ]) = Var(Λ) = 0.5(4)+0.5(4) = 4, since Λ deviates by 2 from mean 4 in each case, giving 4. Total = 4 + 4 = 8. Using only the first term gives 4, a missed between-group variance.
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