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Multinoulli distribution

https://www.statlect.com/probability-distributions/multinoulli-distribution3

 

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 技术分享 outcomes and you denote by 技术分享 a random variable that takes value 1 if you obtain the 技术分享-th outcome and 0 otherwise, then the random vector 技术分享 defined as技术分享is a Multinoulli random vector. In other words, when the 技术分享-th outcome is obtained, the 技术分享-th entry of the Multinoulli random vector 技术分享 takes value 技术分享, while all other entries take value 技术分享.

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

Definition

The distribution is characterized as follows.

Definition Let 技术分享 be a 技术分享 discrete random vector. Let the support of 技术分享 be the set of 技术分享 vectors having one entry equal to 技术分享 and all other entries equal to 技术分享:技术分享Let 技术分享, ..., 技术分享 be 技术分享 strictly positive numbers such that技术分享We say that 技术分享 has a Multinoulli distribution with probabilities 技术分享, ..., 技术分享 if its joint probability mass function is技术分享

If you are puzzled by the above definition of the joint pmf, note that when 技术分享 and 技术分享 because the 技术分享-th outcome has been obtained, then all other entries are equal to 技术分享 and技术分享

Expected value

The expected value of 技术分享 is技术分享where the 技术分享 vector 技术分享 is defined as follows:技术分享

Proof

Covariance matrix

The covariance matrix of 技术分享 is技术分享where 技术分享 is a 技术分享 matrix whose generic entry is技术分享

Proof

Joint moment generating function

The joint moment generating function of 技术分享 is defined for any 技术分享:技术分享

Proof

Joint characteristic function

The joint characteristic function of 技术分享 is技术分享

Proof

Multinoulli distribution