IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
X and Y are independent, with X ~ Poisson(2) and Y ~ Poisson(3). Let S = X + Y. Using MGFs, what is P(S = 2)?
Multiplying the independent Poisson MGFs gives a Poisson MGF with mean 5, so S is Poisson(5). Then P(S=2) = e^-5 × 5²/2 = 0.0842. The value 0.1404 corresponds to P(S=3) or P(S=4), not P(S=2).
- A0.0842Correct
- B0.1404
- C0.0337
- D0.2240
- 0.0067
Explanation
The product of MGFs exp(2(e^t-1))·exp(3(e^t-1)) = exp(5(e^t-1)), so S ~ Poisson(5). P(S=2) = e^-5 × 25/2 = 0.006738 × 12.5 = 0.0842. Option 0.1404 is P(S=3) or P(S=4).
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