IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X ~ Poisson(2) and Y ~ Poisson(3) are independent. Using MGFs, what is P(X + Y = 0)?
The sum of independent Poissons is Poisson with mean 2 + 3 = 5, as the product of MGFs shows. The probability of zero is e^-5. Adding e^-2 and e^-3 is incorrect since independence requires multiplication.
- Ae^-6
- Be^-5Correct
- Ce^-1
- D1 - e^-5
- e^-2 + e^-3
Explanation
The MGF of the sum is exp(2(e^t-1))·exp(3(e^t-1)) = exp(5(e^t-1)), which is Poisson(5). P(0) = e^-5. The option e^-2 + e^-3 wrongly adds the separate probabilities rather than multiplying them.
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