 Probability-generating function

For a nonnegative integer-valued random variable $X$, write $p_k=\mathbb{P}(X=k)$. Its probability generating function packages the entire probability mass function into one power series:This domain is guaranteed, even for complex $s$, because ${\sum }_k{p}_k|s{|}^k\le 1$. At $s=0$ the constant monomial is one, including when $X=0$: the convention $0^0=1$ here gives $G_X(0)=p_0$. Also $G_X(1)=1$. The power series may converge beyond this domain, but that needs an additional argument. Encoding a probability distribution this way turns convolution of independent random variables into multiplication.^[ref]

A probability-generating function (PGF) is a specific type of power series that is used to encode the probabilities of a discrete random variable. It is particularly useful in the study of probability distributions and in solving problems involving sums of independent random variables. ### Definition For a discrete random variable \( X \) that takes non-negative integer values (i.e.