IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X and Y are independent random variables. X has moment generating function M_X(t) = (0.6 + 0.4e^t)^5 and Y has moment generating function M_Y(t) = (0.6 + 0.4e^t)^7. What is the distribution of S = X + Y?
S is Binomial with n = 12 and p = 0.4. The MGF of an independent sum is the product of the individual MGFs, so the exponents 5 and 7 add to 12 while the success probability stays 0.4.
- ABinomial with n = 12 and p = 0.4Correct
- BBinomial with n = 35 and p = 0.4
- CBinomial with n = 12 and p = 0.8
- DPoisson with mean 4.8
- Binomial with n = 2 and p = 0.4
Explanation
For independent variables the MGF of the sum is the product of the MGFs. This gives (0.6 + 0.4e^t)^12, which is the MGF of a Binomial(12, 0.4). Option B wrongly multiplies the n values, and option C wrongly adds the p values.
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