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 
outcomes and you denote by 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 :
Let , ..., be strictly positive numbers such thatWe say that has a Multinoulli distribution with probabilities , ..., if its joint probability mass function is
If you are puzzled by the above definition of the joint pmf, note that when and because the -th outcome has been obtained, then all other entries are equal to and
Expected value
The expected value of iswhere the vector is defined as follows:
Covariance matrix
The covariance matrix of iswhere is a matrix whose generic entry is
Joint moment generating function
The joint moment generating function of is defined for any :
Joint characteristic function
The joint characteristic function of is