Categorical distribution

Categorical
Parameters	$k>0$ number of categories (integer) $p_{1},\ldots ,p_{k}$ event probabilities ( $\Sigma p_{i}=1$ )
Support	$x\in \{1,\dots ,k\}$
pmf	(1) $p(x=i)=p_{i}$ (2) $p(x)=p_{1}^{[x=1]}\cdots p_{k}^{[x=k]}$ (3) $p(x)=[x=1]\cdot p_{1}\,+\dots +\,[x=k]\cdot p_{k}$ $[x=i]$ is the Iverson bracket
CDF	${\begin{cases}0&{\text{for }}x<1\\\sum _{j=1}^{i}p_{j}&{\text{for }}x\in [i,i+1)\\1&{\text{for }}x\geq k\end{cases}}$
Mean	$\operatorname {E} ([x=i])=p_{i}$ , this is the mean of the Iverson bracket $[x=i]$ and not the mean of $x$
Median	$i{\text{ such that }}\sum _{j=1}^{i-1}p_{j}\leq 0.5{\text{ and }}\sum _{j=1}^{i}p_{j}\geq 0.5$
Mode	$i{\text{ such that }}p_{i}=\max(p_{1},\ldots ,p_{k})$
Variance	$\textstyle {\mathrm {Var} }([x=i])=p_{i}(1-p_{i})$ $\textstyle {\mathrm {Cov} }([x=i],[x=j])=-p_{i}p_{j}~~(i\neq j)$
MGF	$\sum _{i=1}^{k}p_{i}e^{t_{i}}$
CF	$\sum _{j=1}^{k}p_{j}e^{it_{j}}$ where $i^{2}=-1$
PGF	$\sum _{i=1}^{k}p_{i}z_{i}{\text{ for }}(z_{1},\ldots ,z_{k})\in \mathbb {C} ^{k}$

(1) $p(x=i)=p_{i}$
(2) $p(x)=p_{1}^{[x=1]}\cdots p_{k}^{[x=k]}$
(3) $p(x)=[x=1]\cdot p_{1}\,+\dots +\,[x=k]\cdot p_{k}$