Proof: Probability mass 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, the probability mass function of $X$ is
\[\label{eq:mult-pmf} f_X(x) = {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i} \; .\]Proof: A multinomial variable is defined as a vector of the numbers of observations belonging to $k$ distinct categories in $n$ independent trials, where each trial has $k$ possible outcomes and the category probabilities are identical across trials.
Since the individual trials are independent and joint probability factorizes under independence, the probability of a particular series of $x_1$ observations for category $1$, $x_2$ observations for category $2$ etc., when order does matter, is
\[\label{eq:mult-prob} \prod_{i=1}^k {p_i}^{x_i} \; .\]When order does not matter, there is a number of series consisting of $x_1$ observations for category $1$, …, $x_k$ observations for category $k$. This number is equal to the number of possibilities in which $x_1$ category $1$ objects, …, $x_k$ category $k$ objects can be distributed in a sequence of $n$ objects which is given by the multinomial coefficient that can be expressed in terms of factorials:
\[\label{eq:mult-coeff} {n \choose {x_1, \ldots, x_k}} = \frac{n!}{x_1! \cdot \ldots \cdot x_k!} \; .\]In order to obtain the probability of $x_1$ observations for category $1$, …, $x_k$ observations for category $k$, when order does not matter, the probability in \eqref{eq:mult-prob} has to be multiplied with the number of possibilities in \eqref{eq:mult-coeff} which gives
\[\label{eq:mult-pmf-qed} p(X=x|n,\left[p_1, \ldots, p_k \right]) = {n \choose {x_1, \ldots, x_k}} \, \prod_{i=1}^k {p_i}^{x_i}\]which is equivalent to the expression above.
Metadata: ID: P99 | shortcut: mult-pmf | author: JoramSoch | date: 2020-05-11, 23:30.