Bivariate Random Variables

A bivariate random variable $(X,Y)$ models two random variables simultaneously when they are related.

The joint CDF for all BRVs is

$F(x,y)\equiv P(X\le x,Y\le y)$

Discrete Bivariate Random Variables

If $X$ and $Y$ are discrete RVs, $(X,Y)$ is a jointly discrete bivariate RV.

• PMF: $f(x,y)=P(X=x,Y=y)$
  • $0\le f(x,y)\le 1$
  • ${\sum }_x{\sum }_{y}f(x,y)=1$
  • $P((X,Y)\in A)=\sum {\sum }_{(x,y)\in A}f(x,y)$

Continuous Bivariate Random Varibales

If $X$ and $Y$ are continuous RVs, $(X,Y)$ is a jointly continuous RV if there exists a joint probability density function (PDF) $f(x,y)$:

• $f(x,y)\ge 0,\forall x,y$
• ${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x,y)dxdy=1$
• $P((X,Y)\in A)={\iint }_{A}f(x,y)dxdy$

$f(x,y)=\frac{{\partial }^2}{\partial x\partial y}F(x,y)$