Categorical distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a <a href="/facts/Discrete_probability_distribution/EpsKKVRu">discrete probability distribution</a> 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 <a href="/facts/Sample_space/WVldkN4F">sample space</a> 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 <a href="/facts/Generalization/v0b1pwUx">generalization</a> of the <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a> for a <a href="/facts/Categorical_variable/3UQNkLqs">categorical</a> random variable, i.e. for a discrete variable with more than two possible outcomes, such as the roll of a <a href="/facts/Dice/qBnylzwJ">die</a>. On the other hand, the categorical distribution is a <a href="/facts/Special_case/bKaJ85XJ">special case</a> of the <a href="/facts/Multinomial_distribution/y58v1p9J">multinomial distribution</a>, in that it gives the probabilities of potential outcomes of a single drawing rather than multiple drawings.

Categorical distribution open-in-new

Categorical distribution