If S = X_1 + ... + X_N with discrete severities independent of N, how do the PGFs of S, N, and X connect?

The aggregate PGF is P_S(z) = P_N(P_X(z)). Conditional on N = n, the severity sum has PGF P_X(z)^n, then averaging over N gives the composition.

N is Poisson(3). Severity X is 1 with probability 0.7 and 2 with probability 0.3. Write P_S(z).

P_X(z) = 0.7z + 0.3z^2, so P_S(z) = exp(3(0.7z + 0.3z^2 - 1)). Steps: Write the severity PGF. Substitute it into the Poisson PGF. Keep the exponent as lambda(P_X(z) - 1).

Before running Panjer recursion for an aggregate distribution, what setup items must be named?

Name the frequency family and its a,b class values, the discrete severity probabilities including f(0), the seed g(0), and the support or truncation range for S.

Concept

Probability Generating Functions

A probability generating function packages a nonnegative-integer distribution into one function. In actuarial aggregate models, PGFs explain why compound distributions compose cleanly and why Poisson, Binomial, and Negative Binomial fit the Panjer framework.

Definition

For a nonnegative-integer random variable N, the probability generating function is P_N(z) = E[z^N]. It is a power series whose coefficient of z^n is the probability P(N = n).

Use PGFs only for count distributions and discrete severity distributions. For continuous severities, use the MGF or Laplace transform, or discretize the severity before applying a PGF or Panjer recursion.

Probability generating function

P_N(z)=E[z^N]=\sum_{n=0}^{\infty}p_n z^n

Recover probabilities

p_n=\frac{1}{n!}P_N^{(n)}(0)

Moments From Derivatives

The first derivative at 1 gives the mean. The second derivative at 1 gives the second factorial moment. Combining those two gives the variance.

This is why PGFs are so efficient for count models: factorial moments often simplify before ordinary moments do.

Mean and variance from PGF

E[N]=P_N'(1),\qquad \operatorname{Var}(N)=P_N''(1)+P_N'(1)-\bigl(P_N'(1)\bigr)^2

Core Frequency PGFs

Poisson is the cleanest case: the PGF is exponential and preserves its form under independent sums. Binomial is finite and polynomial. Negative Binomial has the rational form that makes overdispersed counts tractable.

In the SOA Loss Models parameterization, Negative Binomial uses r and beta, with mean r beta and variance r beta times 1 plus beta. That is the same form used by the gamma-mixed-Poisson identity.

Poisson

P_N(z)=\exp\{\lambda(z-1)\}

Binomial

P_N(z)=(1-q+qz)^m

Negative binomial in SOA form

P_N(z)=\left(1+\beta(1-z)\right)^{-r}

Compound Distributions

Let S = X_1 + ... + X_N, where the X_i are independent nonnegative-integer severities with PGF P_X(z), independent of N. Conditional on N = n, the aggregate PGF is P_X(z)^n. Averaging over N gives the composition rule.

This single identity explains most compound distribution shortcuts. For a compound Poisson, substitute the severity PGF into the Poisson PGF. For compound Binomial or compound Negative Binomial, use the same substitution in the corresponding frequency PGF.

Compound PGF

P_S(z)=P_N(P_X(z))

Compound Poisson PGF

P_S(z)=\exp\{\lambda(P_X(z)-1)\}

MGF parallel

M_S(t)=P_N(M_X(t))

Poisson Sum Example

If N is Poisson with mean 3 and severity X takes values 1 and 2 with probabilities 0.7 and 0.3, then P_X(z) = 0.7z + 0.3z^2. The aggregate PGF is exp(3(0.7z + 0.3z^2 - 1)).

From the derivative rule, E[S] = E[N]E[X] = 3(1.3) = 3.9. The variance is lambda E[X^2] = 3(0.7 + 4(0.3)) = 5.7. The PGF gives the distributional object; the moments fall out as checks.

Panjer Connection

The Panjer recursion is not a random formula. It is the coefficient-recursion consequence of a frequency PGF whose probabilities satisfy p_n = (a + b/n)p_{n-1}. Poisson, Binomial, and Negative Binomial are exactly the core families where this structure is useful.

For ASTAM, the workflow is: identify the frequency family, place or discretize severity on a lattice, determine f(0), calculate the seed probability g(0), and then recurse. The PGF explains why the aggregate PMF exists; the recursion computes it cell by cell.

Zero-Modified Counts

Zero-modified and zero-truncated count models alter the mass at zero while preserving a related positive-count structure. This matters for portfolios where the observed zero frequency is too high or too low for the base model.

The PGF view makes the modification visible: change the zero mass, then rescale the positive part. The mean and variance change because the zero cell is no longer tied to the base model.

Zero-modified PGF shape

P_M(z)=p_0^M+(1-p_0^M)P_T(z)

Practice

Original Source-Backed Practice

4 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.