## Idea A *categorical distribution* is the generic distribution of a random variable with finite image. ## Definition +-- {: .num_defn} ###### Definition A random variable $X$ whose image consists of precisely $n\in \mathbb{N}_0$ 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 $2$. A categorical distribution is precisely a multinomial distribution $B(n,p_1,...,p_k)$ with $n=1$ Categorical distribution is to multinomial distribution like Bernoulli distribution is to binomial distribution. The possible probabilities of a random variable with image-cardinalty $n$ are precisely the points of the standard $(n-1)$-[[nLab:simplex]] embedded into $\mathbb{R}^n$ since we have $1=\sum_{i=0}^{n-1} p_i$.