IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
A claim-count variable N has PGF G(s) = exp(3(s - 1)). Using the PGF, what is Var(N)?
The variance is 3. With G'(1)=3 and G''(1)=9, Var(N)=G''(1)+G'(1)-[G'(1)]^2 = 9+3-9 = 3, consistent with a Poisson distribution of mean 3.
- A9
- B3Correct
- C6
- D1/3
- 12
Explanation
G'(s) = 3G(s), so G'(1) = 3 = E[N]. G''(s) = 9G(s), so G''(1) = 9. Var = G''(1) + G'(1) - G'(1)^2 = 9 + 3 - 9 = 3. Answer 9 wrongly takes G''(1) as the variance.
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