Skip to content

IAI Actuarial Core Principles · Actuarial Statistics · Generating functions

X ~ Poisson(2) and Y ~ Poisson(3) are independent. Using MGFs, which statement about W = X + Y is correct, and what is P(W = 0)?

The product of the MGFs is exp(5(e^t - 1)), so W is Poisson with mean 5. Hence P(W = 0) = e^-5. Multiplying or averaging the means instead of adding them gives wrong answers.

  1. AW is Poisson(5); P(W = 0) = e^-5Correct
  2. BW is Poisson(6); P(W = 0) = e^-6
  3. CW is Poisson(5); P(W = 0) = e^-6
  4. DW is not Poisson; P(W = 0) = e^-5
  5. W is Poisson(2.5); P(W = 0) = e^-2.5

Explanation

M_W(t) = exp(2(e^t-1)) x exp(3(e^t-1)) = exp(5(e^t-1)), the Poisson(5) MGF. So P(W=0) = e^-5. Poisson(6) would multiply the parameters, and 2.5 averages them.

Did you get it right without looking?

One question tells you little. A timed set on Generating functions shows your real accuracy, how long you take and where you lose marks.

Practise Generating functionsFree diagnostic mock

More Generating functions questions