4.5 Probability Generating Functions

Definition

For a discrete random variable $X$ with values $x_i$, the probability generating function is defined as:

$G_X(t) = \sum_x t^x P(X=x) = E(t^X)$

The variable $t$ is called a dummy variable and has no intrinsic meaning, but plays an important role in calculating expectations and variances.

Key Properties

Property 1: When $t = 1$, the PGF equals 1 (since the sum of all probabilities is 1):

$G_X(1) = 1$

Property 2: The coefficient of $t^r$ in the expansion of $G_X(t)$ gives $P(X = r)$.

Property 3: We can extract probabilities from the PGF using:

$P(X=r) = \frac{G_X^{(r)}(0)}{r!}$

where $G_X^{(r)}(0)$ denotes the $r$-th derivative of $G_X(t)$ evaluated at $t = 0$.

Constructing PGFs from Probability Distributions

Example 1: PGF from a Probability Table

Consider the following probability distribution:

The PGF is constructed by applying the definition:

$G_X(t) = 0.1t^0 + 0.2t^1 + 0.3t^2 + 0.15t^3 + 0.1t^4 + 0.1t^5 + 0.05t^6$

$G_X(t) = 0.1 + 0.2t + 0.3t^2 + 0.15t^3 + 0.1t^4 + 0.1t^5 + 0.05t^6$

Example 2: Non-consecutive Values

For a distribution with $x = 2, 4, 5, 10$ and probabilities $0.1, 0.2, 0.3, 0.4$ respectively:

$G_X(t) = 0.1t^2 + 0.2t^4 + 0.3t^5 + 0.4t^{10}$

Notice that the powers of $t$ correspond to the values of $x$, not sequential indices.