Trinomial Distribution

Definition

$X\sim \mathrm{Tri}(n,p_1,p_2)$ where $n$ is the number of trials, $p_1$ is a probability of category $1$, and $p_2$ is a probability of category $2$

Distribution that expands the outcomes of a binomial distribution to three

Properties

PDF

$f(x,y)=\frac{n!}{x!y!(n-x-y)!}{p}_1^x{p}_2^y(1-p_1-p_2{)}^{n-x-y}$

MGF

$M_{X,Y}(t_1,t_2)=(p_1{e}^{t_1}+p_2{e}^{t_2}+p_3{)}^n$

Marginal PDF

$f_X(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p_1^x(1-p_1{)}^{n-x}\sim B(n,p_1)$

$f_Y(y)\sim B(n,p_2)$

Conditional PDF

$f_{Y|X}(y|x)=(\genfrac{}{}{0ex}{}{n-x}{y}){\left(\frac{p_2}{1-p_1}\right)}^y{\left(1-\frac{p_2}{1-p_1}\right)}^{n-x-y}\sim B\left(n-x,\frac{p_2}{1-p_1}\right)$

$f_{X|Y}(x|y)\sim B\left(n-y,\frac{p_1}{1-p_2}\right)$