IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
N has PGF G(s) = s/(2 - s) for |s| < 2. What is E[N(N - 1)]?
E[N(N-1)] equals the second derivative of the PGF at s=1. Differentiating s/(2-s) twice gives 4/(2-s)^3, which equals 4 at s=1.
- A4Correct
- B2
- C6
- D8
- 1
Explanation
N is geometric with p = 1/2 on {1,2,...}. G'(s) = 2/(2-s)^2, G''(s) = 4/(2-s)^3. At s=1, G''(1) = 4, and E[N(N-1)] = G''(1) = 4. Answer 2 wrongly uses G'(1).
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