Bivariate continuous trinomial distribution

Model setting

$f(x,y)=C \times λ_{1}^x \times λ_{2}^y \times (1-λ_{1}-λ_{2})^{1-x-y}$

where $0 < x < 1, 0 < y < 1, 0 < x + y < 1$ .

This distribution has two parameters, $λ_{1}, λ_{2}$, and the parameter space are $0<λ_{1}<1, 0<λ_{2}<1$.

Let $f(x)=\int f(x,y)dy$. and $X$ is not Continuous Bernoulli distribution( $λ_{1}$ ).

Let $f(y)=\int f(x,y)dx$ , and $Y$ is not Continuous Bernoulli distribution( $λ_{2}$ ).

The marginal probability density function of $X$ and $Y$ have two parameters, $λ_{1}, λ_{2}$ and $\int f(x)dx= \int f(y)dy=1$.

$X$ and $Y$ are not independent random variables.

$X+Y$ is not Continuous Bernoulli distribution ( $λ_{1}, λ_{2}$ ).

The $(x,y,f(x,y))$ dynamic diagrams is affected by $λ_{1}$ and setting $λ_{1}+λ_{2}=c$.

This displayed method can understand $f(x,y)$ diagram changed when the $λ_{1}$ different value in simply.

$f(x) = C \times λ^{x} \times (1 - λ)^{1 - x}$, $0 < x < 1$ ,

$\int f(x)dx = 1$.

Let $λ =0.01$ to $0.99$ and step $= 0.01$ .

Video

$λ_{1} + λ_{2} = 0.1$ , $λ_{1} = 0.01$ to $0.099$ and step $= 0.001$

Video for case 1

Video for case 1 of X

Video for case 1 of Y