Index: The Book of Statistical Proofs ▷ Probability 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

\[\label{eq:pgf} G_X(z) = \sum_{x=0}^{\infty} f_X(x) \, z^x\]

With the probability mass function of the binomial distribution

\[\label{eq:bin-pmf} f_X(x) = {n \choose x} \, p^x \, (1-p)^{n-x} \; ,\]

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}.

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