Multinoulli distribution

The Multinoulli distribution (sometimes also called categorical distribution) is a generalization of the Bernoulli distribution. If you perform an experiment that can have only two outcomes (either success or failure), then a random variable that takes value 1 in case of success and value 0 in case of failure is a Bernoulli random variable. If you perform an experiment that can have outcomes and you denote by $X_{i}$ a random variable that takes value 1 if you obtain the -th outcome and 0 otherwise, then the random vector defined asis a Multinoulli random vector. In other words, when the -th outcome is obtained, the -th entry of the Multinoulli random vector takes value , while all other entries take value .

In what follows the probabilities of the possible outcomes will be denoted by .

Definition

The distribution is characterized as follows.

Definition Let be a discrete random vector. Let the support of be the set of vectors having one entry equal to and all other entries equal to : [eq3] Let $p_{1}$ , ..., $p_{K}$ be strictly positive numbers such that [eq4] We say that has a Multinoulli distribution with probabilities $p_{1}$ , ..., $p_{K}$ if its joint probability mass function is [eq5]

If you are puzzled by the above definition of the joint pmf, note that when and $x_{i}=1$ because the -th outcome has been obtained, then all other entries are equal to and [eq7]

Expected value

The expected value of iswhere the vector is defined as follows:

Proof

Covariance matrix

The covariance matrix of iswhere is a matrix whose generic entry is [eq12]

Proof

Joint moment generating function

The joint moment generating function of is defined for any $tin U{211d} ^{K}$ : [eq16]

Proof

Joint characteristic function

The joint characteristic function of is [eq19]

Proof