IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
A random variable X has moment generating function M(t) = exp(3t + 2t^2) for all real t. What is the cumulant generating function of X, and what is Var(X)?
The cumulant generating function is the natural log of the MGF, giving 3t + 2t^2. The variance is the second derivative at zero, which is 4. This is a normal distribution with mean 3 and variance 4, so the coefficient 2 is half the variance.
- AK(t) = 3t + 2t^2; Var(X) = 2
- BK(t) = 3t + 2t^2; Var(X) = 4Correct
- CK(t) = exp(3t + 2t^2); Var(X) = 4
- DK(t) = 3t + 2t^2; Var(X) = 13
- K(t) = 3 + 4t; Var(X) = 4
Explanation
K(t) = ln M(t) = 3t + 2t^2. The variance is K''(0) = 4. Choosing 2 is the mistake of reading the coefficient of t^2 as the variance instead of doubling it (this is the N(3,4) case). Option 13 is E[X^2] = 4 + 9, not the variance.
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