IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X is Poisson with mean 4, so its cumulant generating function is K_X(t) = 4(e^t - 1). Y = 3X + 2. What is the third cumulant of Y?
The third cumulant of Y is 108. Cumulants of order r scale by a^r under Y = aX + b, and the shift does not affect orders above one. A Poisson variable with mean 4 has third cumulant 4, and 3^3 × 4 = 108.
- A108Correct
- B12
- C4
- D27
- 36
Explanation
For Y = aX + b, K_Y(t) = bt + K_X(at), so the third cumulant scales by a^3 and the shift b has no effect on it. Every cumulant of a Poisson variable equals its mean, so the third cumulant of X is 4. Then 27 × 4 = 108. The value 36 comes from wrongly scaling by a^2 (9 × 4).
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