IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
Claim sizes X (in Rs thousand) follow an exponential distribution with mean 4, so M(t) = 1/(1 - 4t) for t < 0.25. Using the MGF, what is E[X^3]?
For an exponential with mean 4, the third moment is 3! x 4^3 = 6 x 64 = 384. This follows from differentiating 1/(1 - 4t) three times at t = 0. Omitting the factorial gives 64, which is wrong.
- A64
- B128
- C384Correct
- D24
- 256
Explanation
E[X^n] = n! x 4^n for an exponential with mean 4. For n = 3: 6 x 64 = 384. Answer 64 forgets the factorial, and 256 is 4^4.
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