Categorical distribution ($\mathrm{Cat}(p_1,\dots ,p_k)$)

A categorical distribution describes one trial with one of $k$ labelled outcomes and probabilities $p_1,\dots ,p_k$ summing to one. The labels need not be numbers or have an order. It is the single-trial building block of the multinomial distribution: $n$ independent draws with the same categorical distribution produce multinomial category counts.

For a trial choosing labels $A,B,C$ with probabilities $1/2,3/10,1/5$, the category-indicator random vector is $(1,0,0)$, $(0,1,0)$, or $(0,0,1)$. Its expected value is $(1/2,3/10,1/5)$; its components are dependent because exactly one is one. A numerical encoding of $A,B,C$ is a separate modelling choice: the category labels themselves do not define an arithmetic mean.