Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate discrete distributions ▷ Binomial distribution ▷ Probability-generating function

Theorem: Let $X$ be a random variable following a binomial distribution:

$\label{eq:bin} X \sim \mathrm{Bin}(n,p) \; .$

Then, the probability-generating function of $X$ is

$\label{eq:bin-pgf} G_X(z) = (q + pz)^n$

where $q = 1-p$.

Proof: The probability-generating function of $X$ is defined as

$\label{eq:pgf} G_X(z) = \sum_{x=0}^{\infty} f_X(x) \, z^x$ $\label{eq:bin-pmf} f_X(x) = {n \choose x} \, p^x \, (1-p)^{n-x} \; ,$

we obtain:

$\label{eq:pgf-zero-s1} \begin{split} G_X(z) &= \sum_{x=0}^{n} {n \choose x} \, p^x \, (1-p)^{n-x} \, z^x \\ &= \sum_{x=0}^{n} {n \choose x} \, (pz)^x \, (1-p)^{n-x} \; . \end{split}$

According to the binomial theorem

$\label{eq:bin-th} (x+y)^n = \sum_{k=0}^{n} {n \choose k} \, x^{n-k} \, y^k \; ,$

the sum in equation \eqref{eq:pgf-zero-s1} equals

$\label{eq:pgf-zero-s2} G_X(z) = \left( (1-p) + (pz) \right)^n$

which is equivalent to the result in \eqref{eq:bin-pgf}.

Sources:

Metadata: ID: P363 | shortcut: bin-pgf | author: JoramSoch | date: 2022-10-11, 09:25.