nLab
categorical distribution

Idea

A categorical distribution is the generic distribution of a random variable with finite image.

Definition

Definition

A random variable XX whose image consists of precisely n 0n\in \mathbb{N}_0 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 22.

A categorical distribution is precisely a multinomial distribution B(n,p 1,...,p k)B(n,p_1,...,p_k) with n=1n=1

Categorical distribution is to multinomial distribution like Bernoulli distribution is to binomial distribution.

The possible probabilities of a random variable with image-cardinalty nn are precisely the points of the standard (n1)(n-1)-simplex embedded into n\mathbb{R}^n since we have 1= i=0 n1p i1=\sum_{i=0}^{n-1} p_i. From this viewpoint we see that for a time-homogenous Markov-chain with finite state space the exists a stationary distribution π\pi which is a fixed point of the linear (hence continuous) transformation on the unit simplex associated to the transition matrix PP of the chain since any continuous transformation in the unit simplex has a fixed point.

References

Last revised on October 23, 2012 at 19:00:17. See the history of this page for a list of all contributions to it.