Proof: Cumulative distribution function of the multinomial distribution
Theorem: Let $X$ be a random vector following a multinomial distribution:
\[\label{eq:mult} X \sim \mathrm{Mult}(n,\left[p_1, \ldots, p_k \right]) \; .\]Then, 1) the joint cumulative distribution function of $X$ is
\[\label{eq:mult-cdf-joint} F_X(x) = {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i}\]and 2) the marginal cumulative distribution function of each $X_i$ is
\[\label{eq:mult-cdf-marg} F_{X_i}(x_i) = \sum_{k=0}^{x_i} {n \choose k} \, p_i^k \, (1-p_i)^{n-k} \; .\]Proof: The probability mass function of the multinomial distribution is
\[\label{eq:mult-pmf} f_X(x) = {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i} \; ,\]1) The joint cumulative distribution function (CDF) of a random vector is defined as
\[\label{eq:cdf-joint} F_X(x) = \mathrm{Pr}(X_1 \leq x_1, \ldots, X_n \leq x_n)\]which, in terms of the probability mass function, is given by
\[\label{eq:cdf-pmf} F_X(x) = \sum_{t_1 \leq x_1} \ldots \sum_{t_n \leq x_n} f_X(t_1, \ldots, t_n) \; .\]For a given set of counts $x_1, \ldots, x_k$ satisfying $\sum_{i=1}^k x_i = n$, we define the set of eligible values $t_1, \ldots, t_k$ for the CDF:
\[\label{eq:ti-set} \mathcal{T}(x) = \left\lbrace (t_1, \ldots, t_k) \mid t_i \in \mathbb{N}_0, \; 0 \leq t_i \leq x_i, \; \sum_{i=1}^k t_i = n \right\rbrace \; .\]Since any eligible set of counts $t_1, \ldots, t_k$ must add up to $n$, this set is equivalent to
\[\label{eq:ti-set-eq} \mathcal{T}(x) = \left\lbrace (x_1, \ldots, x_k) \right\rbrace \; .\]Therefore, the CDF is equal to the probability mass function at $t = x$:
\[\label{eq:mult-cdf-joint-qed} F_X(x) \overset{\eqref{eq:cdf-pmf}}{=} \sum_{t \in \mathcal{T}(x)} f_X(t_1, \ldots, t_k) = f_X(x_1, \ldots, x_k) \; .\]so that the joint cumulative distribution function of the multinomial distribution becomes
\[\label{eq:xi-ni-con} F_X(x) \overset{\eqref{eq:mult-pmf}}{=} {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i} \; .\]2) The marginal distributions for the multinomial distribution are
\[\label{eq:mult-marg} X_i \sim \mathrm{Bin}(n, p_i) \quad \text{for all} \quad i = 1, \ldots, k \; .\]Thus, the marginal cumulative distribution function of any entry $X_i$ is equal to the cumulative distribution function of the binomial distribution:
\[\label{eq:mult-cdf-marg-qed} F_{X_i}(x_i) = \sum_{k=0}^{x_i} {n \choose k} \, p_i^k \, (1-p_i)^{n-k} \; .\]Metadata: ID: P489 | shortcut: mult-cdf | author: JoramSoch | date: 2025-02-06, 15:15.