Probability Generating Functions

Probability · Axiom Academy

LESSON Probability Generating Functions Transform discrete random variables into power series to extract probabilities and moments For a discrete random variable X taking non-negative integer values, the probability generating function is defined as: Each coefficient of s^k in the expansion corresponds to P(X = k). The animation shows how the PGF encodes the entire probability distribution. We can recover individual probabilities from the PGF by evaluating derivatives at s = 0: The k-th derivative at s = 0, divided by k!, gives us P(X = k). This follows from the power series expansion where the k-th term is P(X = k)s^k. 3. Computing Moments from the PGF The PGF provides a systematic way to compute moments. The mean and variance can be extracted using derivatives: The first derivative at s = 1 gives the mean E[X]. The variance can be found using E[X(X-1)] = G''(1), which leads to Var(X) = G''(1) + G'(1) - [G'(1)]^2. One of the most powerful properties: if X and Y are independent, then the PGF of their sum is the product of their individual PGFs: This multiplication property makes PGFs especially useful for analyzing sums of independent random variables, such as in branching processes or compound distributions. 5. Example: Geometric Distribution Consider a geometric random variable X with parameter p (number of trials until first success). Its PGF is:

This is the written version of the interactive lesson above. See the full Probability course.