IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X has MGF M(t) = exp(2t + 3t^2). What is E[X^2]?
E[X^2] equals 10. The MGF exp(2t + 3t^2) is Normal with mean 2 and variance 6, so the second moment is variance plus mean squared, which is 6 + 4 = 10.
- A6
- B10Correct
- C4
- D16
- 12
Explanation
This is the MGF of a Normal distribution with mean 2 and variance 6 (exp(μt + σ²t²/2), so σ² = 6). E[X^2] = variance + mean^2 = 6 + 4 = 10. The value 6 is only the variance.
Did you get it right without looking?
One question tells you little. A timed set on Generating functions shows your real accuracy, how long you take and where you lose marks.
More Generating functions questions
- X has MGF M(t) = (1 − 2t)^-3 for t < 0.5. What is the mean of X?
- Claims counts N follow a Poisson distribution with mean 4. The cumulant generating function is K(t) = 4(e^t - 1). What is the third central …
- X1, X2, X3 are independent Exponential random variables each with mean 2. What is the moment generating function of their sum, valid for t <…
- X ~ Poisson(2) and Y ~ Poisson(3) are independent. Using MGFs, which statement about W = X + Y is correct, and what is P(W = 0)?
- N has PGF G(s) = s/(2 - s) for |s| < 2. What is E[N(N - 1)]?
- X has MGF M(t) = 1/(1 - 2t) for t < 1/2. What is E[X³]?