Multinomial distribution

Multinomial

Parameters	${\displaystyle n>0}$ number of trials (integer) ${\displaystyle p_{1},\ldots ,p_{k}}$ event probabilities (${\displaystyle \Sigma p_{i}=1}$)
Support	${\displaystyle X_{i}\in \{0,\dots ,n\}}$ ${\displaystyle \Sigma X_{i}=n\!}$
pmf	${\displaystyle {\frac {n!}{x_{1}!\cdots x_{k}!}}p_{1}^{x_{1}}\cdots p_{k}^{x_{k}}}$
Mean	${\displaystyle E\{X_{i}\}=np_{i}}$
Variance	${\displaystyle \textstyle {\mathrm {Var} }(X_{i})=np_{i}(1-p_{i})}$ ${\displaystyle \textstyle {\mathrm {Cov} }(X_{i},X_{j})=-np_{i}p_{j}~~(i\neq j)}$
MGF	${\displaystyle {\biggl (}\sum _{i=1}^{k}p_{i}e^{t_{i}}{\biggr )}^{n}}$
CF	${\displaystyle \left(\sum _{j=1}^{k}p_{j}e^{it_{j}}\right)^{n}}$ where ${\displaystyle i^{2}=-1}$
PGF	${\displaystyle {\biggl (}\sum _{i=1}^{k}p_{i}z_{i}{\biggr )}^{n}{\text{ for }}(z_{1},\ldots ,z_{k})\in \mathbb {C} ^{k}}$

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for rolling a k-sided dice n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

When n is 1 and k is 2 the multinomial distribution is the Bernoulli distribution. When k is 2 and number of trials are more than 1 it is the binomial distribution. When n is 1 it is the categorical distribution.

The Bernoulli distribution is the probability distribution of whether a Bernoulli trial is a success. In other words, it models the number of heads from flipping a (possibly biased) coin one time. The binomial distribution generalizes this to the number of heads from doing n independent flips of the same coin. For the multinomial distribution the analog to the Bernoulli Distribution is the categorical distribution. Instead of flipping one coin, the categorical distribution models the roll of one k sided die. So the multinomial distribution can model n independent rolls of a k sided die.

Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p₁, ..., p_k, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have p_i ≥ 0 for i = 1, ..., k and ${\displaystyle \sum _{i=1}^{k}p_{i}=1}$. Then if the random variables X_i indicate the number of times outcome number i is observed over the n trials, the vector X = (X₁, ..., X_k) follows a multinomial distribution with parameters n and p, where p = (p₁, ..., p_k). While the trials are independent, their outcomes X are dependent because they must be summed to n.

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