IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X and Y are independent, X ~ Poisson(2) and Y ~ Poisson(3). Let S = X + Y. Using MGFs, what is P(S = 0)?
P(S = 0) = e^-5. Multiplying the independent Poisson MGFs gives a Poisson MGF with parameter 2 + 3 = 5, so the probability of zero is e^-5.
- Ae^-2
- Be^-3
- Ce^-6
- De^-5Correct
- e^-1
Explanation
The MGF of S is the product exp(2(e^t-1)) × exp(3(e^t-1)) = exp(5(e^t-1)), so S ~ Poisson(5). P(S=0) = e^-5. Using e^-6 would wrongly multiply the means.
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