Multinomial distribution information

Multinomial
Parameters	number of trials (integer); number of mutually exclusive events (integer); event probabilities, where
Support
PMF
Mean
Variance	;
Entropy
MGF
CF	where
PGF

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided dice rolled 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 k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial distribution. When k is bigger than 2 and n is 1, it is the categorical distribution. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so n determines the suffix, and k the prefix).

The Bernoulli distribution models the outcome of a single Bernoulli trial. In other words, it models whether flipping a (possibly biased) coin one time will result in either a success (obtaining a head) or failure (obtaining a tail). The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided die n times.

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 $\sum _{i=1}^{k}p_{i}=1$ . 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_i are dependent because they must be summed to n.

Multinomial distribution information

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Multinomial distribution

Dirichlet distribution

Categorical distribution

Negative multinomial distribution

Multinomial theorem

Dirichlet negative multinomial distribution

Multinomial

Joint probability distribution

List of probability distributions

Poisson distribution

Hypergeometric distribution

Probability distribution

Binomial distribution

Generalized linear model

Negative binomial distribution

Posterior predictive distribution

Beta distribution

Gibbs sampling

Multinomial logistic regression

Compound probability distribution

Boltzmann distribution

Gumbel distribution

Latent variable model

Poisson binomial distribution

Probability mass function

Exponential family

Parameters	$n>0$ number of trials (integer) $k>0$ number of mutually exclusive events (integer) $p_{1},\ldots ,p_{k}$ event probabilities, where $p_{1}+\dots +p_{k}=1$
Support	$\left\lbrace (x_{1},\dots ,x_{k})\,{\Big \vert }\,\sum _{i=1}^{k}x_{i}=n,x_{i}\geq 0\ (i=1,\dots ,k)\right\rbrace$
PMF	${\frac {n!}{x_{1}!\cdots x_{k}!}}p_{1}^{x_{1}}\cdots p_{k}^{x_{k}}$
Mean	$\operatorname {E} (X_{i})=np_{i}$
Variance	$\operatorname {Var} (X_{i})=np_{i}(1-p_{i})$ $\operatorname {Cov} (X_{i},X_{j})=-np_{i}p_{j}~~(i\neq j)$
Entropy	$-\log(n!)-n\sum _{i=1}^{k}p_{i}\log(p_{i})+\sum _{i=1}^{k}\sum _{x_{i}=0}^{n}{\binom {n}{x_{i}}}p_{i}^{x_{i}}(1-p_{i})^{n-x_{i}}\log(x_{i}!)$
MGF	${\biggl (}\sum _{i=1}^{k}p_{i}e^{t_{i}}{\biggr )}^{n}$
CF	$\left(\sum _{j=1}^{k}p_{j}e^{it_{j}}\right)^{n}$ where $i^{2}=-1$
PGF	${\biggl (}\sum _{i=1}^{k}p_{i}z_{i}{\biggr )}^{n}{\text{ for }}(z_{1},\ldots ,z_{k})\in \mathbb {C} ^{k}$