IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
The number of policies N sold by an agent has PGF G_N(s) = s^2. Each policy independently earns a bonus-count X with PGF G_X(s) = 0.5 + 0.5s. What is the PGF of the total S = X_1 + ... + X_N?
The PGF of a random sum is G_N evaluated at G_X(s). With G_N(s)=s squared, this gives (0.5+0.5s) squared, the Binomial(2, 0.5) PGF.
- A(0.5 + 0.5s)^2Correct
- B0.5 + 0.5s^2
- C(0.5 + 0.5s)^4
- D0.25 + 0.5s + 0.25s^3
- 0.5s^2
Explanation
The compound PGF is G_N(G_X(s)) = (0.5+0.5s)^2. Since N is fixed at 2, S is the sum of two Bernoulli(0.5) variables. Option 0.5+0.5s^2 wrongly composes in the reverse order.
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