बहुपद वितरण

Multinomial

Parameters	$n>0$ number of trials (integer) $k>0$ number of mutually exclusive events (integer) $p_{1},\ldots ,p_{k}$ event probabilities, where $p_{1}+\dots +p_{k}=1$
Support	$\left\lbrace (x_{1},\dots ,x_{k})\,{\Big \vert }\,\sum _{i=1}^{k}x_{i}=n,x_{i}>0(i=1,\dots ,k)\right\rbrace$
PMF	${\frac {n!}{x_{1}!\cdots x_{k}!}}p_{1}^{x_{1}}\cdots p_{k}^{x_{k}}$
Mean	$\operatorname {E} (X_{i})=np_{i}$
Variance	$\operatorname {Var} (X_{i})=np_{i}(1-p_{i})$ $\operatorname {Cov} (X_{i},X_{j})=-np_{i}p_{j}~~(i\neq j)$
Entropy	$-<!--\; -\; -->\log <!--\; \; -->(n!)-<!--\; -\; -->n\underset{i=1}{\overset{k}{\sum <!--\; \sum \; -->}}{p}_i\log <!--\; \; -->(p_i)+\underset{i=1}{\overset{k}{\sum <!--\; \sum \; -->}}\underset{x_i=0}{\overset{n}{\sum <!--\; \sum \; -->}}(\genfrac{}{}{0ex}{}{n}{x_i})p_i^{x_i}{(1-<!--\; -\; -->p_i)}^{n-<!--\; -\; -->x_i}\log <!--\; \; -->({}_{}$