IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X has MGF M(t) = 1/(1 - 2t) for t < 1/2. What is E[X³]?
The MGF is that of an exponential with mean 2, so E[X^k] = k! × 2^k. For k = 3 this gives 6 × 8 = 48. Answering 8 would omit the factorial term from the Taylor expansion.
- A8
- B12
- C48Correct
- D24
- 6
Explanation
M(t) = Σ (2t)^k, so the coefficient of t³ is 8 and E[X³] = 3! × 8 = 48. This is an exponential with mean 2, and E[X^k] = k! 2^k. The value 8 forgets the factorial 3!.
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