A categorical distribution is the generic distribution of a random variable with finite image.
A random variable whose image consists of precisely elements is called to obey a categorical distribution.
By definition every random variable with finite image is categorically distributed.
A Bernoulli distribution is precisely a categorical distribution of a random variable whose image-size is .
A categorical distribution is precisely a multinomial distribution with
Categorical distribution is to multinomial distribution like Bernoulli distribution is to binomial distribution.
The possible probabilities of a random variable with image-cardinalty are precisely the points of the standard -simplex embedded into since we have . From this viewpoint we see that for a time-homogenous Markov-chain with finite state space the exists a stationary distribution which is a fixed point of the linear (hence continuous) transformation on the unit simplex associated to the transition matrix of the chain since any continuous transformation in the unit simplex has a fixed point.
Last revised on October 23, 2012 at 19:00:17. See the history of this page for a list of all contributions to it.