Probability-generating function

A probability-generating function is the power series G(z)=E[z^X] whose coefficient of z^k equals the probability that a nonnegative integer-valued random variable X equals k.

Probability mass function: A function assigning each possible value of a discrete random variable its probability. Its probabilities become the coefficients of the generating function.
Discrete random variable: A random variable whose possible values form a finite or countable set. The function is most directly defined for nonnegative integer-valued discrete variables.
Convolution (probability): The operation that gives the distribution of a sum of independent random variables from their individual distributions. Multiplying generating functions performs the same sum-distribution calculation through coefficients.
Characteristic function (probability theory): The expectation E[e^{itX}], defined for every real-valued random variable. Unlike this function, it always exists and handles variables that are not nonnegative integers.
Probability-generating function of a sum: The generating function describing the sum of independent nonnegative integer-valued random variables. Independence turns the sum's generating function into a product of the individual functions.
Power series: An infinite series of powers of a variable, usually considered within its radius of convergence. A probability-generating function is a power series with probability coefficients.
Branching process: A stochastic process in which individuals independently produce random numbers of offspring. The offspring generating function yields extinction probabilities through a fixed-point equation.
Cumulant-generating function: The logarithm of a moment-generating function, when defined near zero, encoding cumulants. It organizes dependence and cumulants rather than listing discrete probabilities as coefficients.
Factorial moment: The expectation of a falling factorial of a random variable, such as E[X(X−1)]. Derivatives at z=1 yield factorial moments directly.
Expectation: The probability-weighted average value of a random variable. Writing the function as E[z^X] defines its coefficients through probability-weighted outcomes.

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