IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X1, X2, X3 are independent Exponential random variables each with mean 2. What is the moment generating function of their sum, valid for t < 0.5?
The MGF is (1 − 2t)^(−3). Each exponential with mean 2 has MGF (1 − 2t)^(−1), and multiplying three independent copies gives the cube, which is a Gamma(3, rate 0.5) distribution.
- A(1 − 2t)^(−3)Correct
- B(1 − 2t)^(−1)
- C(1 − 6t)^(−1)
- D(1 − 0.5t)^(−3)
- (1 − 2t)^3
Explanation
Each exponential with mean 2 has rate 0.5 and MGF 0.5/(0.5 − t) = (1 − 2t)^(−1). The MGF of the independent sum is the cube of this, (1 − 2t)^(−3), a Gamma(3, 0.5). Option C wrongly adds the means inside one exponential.
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