Trinomial Distribution

Definition

$\mathbf{X}\sim \text{Trinomial}(n;p_1,p_2,p_3)$

Like binomial, but with 3 possible outcomes. Special case of multinomial with $k=3$.

• pmf: $\frac{n!}{x_1!x_2!x_3!}{p}_1^{x_1}{p}_2^{x_2}{p}_3^{x_3}$, where $x_1+x_2+x_3=n$
• mean: $E(X_i)=n{p}_i$
• var: $\text{Var}(X_i)=n{p}_i(1-p_i)$
• mgf: $(p_1{e}^{t_1}+p_2{e}^{t_2}+p_3{e}^{t_3}{)}^n$

Conditional distribution: $E(X_1|X_2=x_2)=(n-x_2)\left(\frac{p_1}{1-p_2}\right)$

Correlation coefficient: $\rho (X_1,X_2)=-\sqrt{\frac{p_1{p}_2}{(1-p_1)(1-p_2)}}$