Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate discrete distributions ▷ Binomial distribution ▷ Special case of multinomial distribution

Theorem: The binomial distribution with sample size $n$ and success probability $p$ is a special case of the multinomial distribution with sample size $n$ category probabilities $p_1 = p$ and $p_2 = 1-p$:

\[\label{eq:bin-mult} X \sim \mathrm{Mult}(n, \left[ p, 1-p \right]) \quad \Rightarrow \quad X \sim \mathrm{Bin}(p) \; .\]

Proof: The probability mass function of the multinomial distribution, where $x$ is a $1 \times k$ vector with $x_i \in \left\lbrace 0, 1, \ldots, n \right\rbrace$, is as follows:

\[\label{eq:mult-pmf} \mathrm{Mult}(x; n, \left[ p_1, \ldots, p_k \right]) = {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i} \; .\]

If we let $k = 2$, $p_1 = p$ and $p_2 = 1-p$, we obtain

\[\label{eq:mult-pmf-bin-s1} \mathrm{Mult}(x; n, \left[ p, 1-p \right]) = {n \choose {x_1, x_2}} \, p^{x_1} \, (1-p)^{n-x_1} \; .\]

Recognizing that $\sum_{i=1}^{k} x_i = n$, such that $x_2 = n - x_1$, we get

\[\label{eq:mult-pmf-bin-s2} \begin{split} \mathrm{Mult}(x; n, \left[ p, 1-p \right]) &= {n \choose {x_1, n-x_1}} \, p^{x_1} \, (1-p)^{n-x_1} \\ &= {n \choose x_1} \, p^{x_1} \, (1-p)^{n-x_1} \end{split}\]

which is equivalent to the probability mass function of the binomial distribution.

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Metadata: ID: P495 | shortcut: bin-mult | author: JoramSoch | date: 2025-04-04, 14:40.