Categorical distribution

Categorical
Parameters	number of categories (integer); event probabilities
Support
PMF	(1) ; (2) ; (3) where is the Iverson bracket
Mode

In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution^[1]) is a discrete probability distribution that describes the possible results of a random variable that can take on one of K possible categories, with the probability of each category separately specified. There is no innate underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution, (e.g. 1 to K). The K-dimensional categorical distribution is the most general distribution over a K-way event; any other discrete distribution over a size-K sample space is a special case. The parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1.

The categorical distribution is the generalization of the Bernoulli distribution for a categorical random variable, i.e. for a discrete variable with more than two possible outcomes, such as the roll of a die. On the other hand, the categorical distribution is a special case of the multinomial distribution, in that it gives the probabilities of potential outcomes of a single drawing rather than multiple drawings.

^ Murphy, K. P. (2012). Machine learning: a probabilistic perspective, p. 35. MIT press. ISBN 0262018020.

[1] Murphy, K. P. (2012). Machine learning: a probabilistic perspective, p. 35. MIT press. ISBN 0262018020.

[1]

Parameters	$k>0$ number of categories (integer) $p_{1},\ldots ,p_{k}$ event probabilities $(p_{i}\geq 0,\,\Sigma p_{i}=1)$
Support	$x\in \{1,\dots ,k\}$
PMF	(1) $p(x=i)=p_{i}$ (2) $p(x)=p_{1}^{[x=1]}\cdots p_{k}^{[x=k]}$ (3) $p(x)=[x=1]\cdot p_{1}\,+\cdots +\,[x=k]\cdot p_{k}$ where $[x=i]$ is the Iverson bracket
Mode	$i{\text{ such that }}p_{i}=\max(p_{1},\ldots ,p_{k})$