IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X has MGF M(t) = (1 - 2t)^(-3), t < 0.5, and Y = 2X + 1. What is the variance of Y?
X is gamma with shape 3 and scale 2, so Var(X) = 3 x 4 = 12. Then Var(2X + 1) = 4 x 12 = 48. Option 96 is wrong and 12 forgets the multiplier.
- A24
- B48
- C96Correct
- D12
- 49
Explanation
X is Gamma(alpha = 3, rate 0.5), scale 2. Var(X) = alpha x scale^2 = 3 x 4 = 12. Var(Y) = 4 x 12 = 48. Check via cumulants: ln M = -3 ln(1-2t), second derivative at 0 = 3 x 4 = 12. So Var(Y) = 48.
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