IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X and Y are independent. X is Normal with mean 2 and variance 9. Y has cumulant generating function K_Y(t) = 5(e^t - 1). What is the third cumulant of X + Y?
The third cumulant of X + Y is 5. Cumulants add for independent variables. The Normal variable contributes zero to the third cumulant, and the Poisson variable with mean 5 contributes 5. The sum is therefore 5.
- A0
- B5Correct
- C14
- D9
- 19
Explanation
For independent variables the cumulant generating functions add, so every cumulant of X + Y is the sum of the individual cumulants. A Normal variable has zero cumulants beyond the second, so its third cumulant is 0. Y is Poisson with mean 5, so its third cumulant is 5. The total is 0 + 5 = 5. The value 14 would be the sum of the two means and variances mixed up, and 19 adds the wrong terms.
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