Joint Probability Distribution Function

If (X,Y) is a two dimensional random continuous variable such that

$$P\left\{ x-\frac{dx}{2}\le x\le x+\frac{dx}{2},y-\frac{dy}{2}\le y\le y+\frac{dy}{2} \right\}=f(x,y)dxdy$$  

Then f(x,y) is called the joint probability density function of (X,Y) provided f(x,y) satisfies the following conditions

(1) $f(x,y)\ge 0,(x,y)\in R$ where R is the range space.

(2) $\iint_{R}{f(x,y)dxdy=1}$  

If $D$ is a subspace of the range space R, $P\{(X,Y)\in D\}$ is defined as,

$$P\{(X,Y)\in D\}=\iint\limits_{D}{f(x,y)dxdy}$$  

In particular, $P(a\le X\le b,c\le Y\le d\}=\int\limits_{c}^{d}{\int\limits_{a}^{b}{f(x,y)dxdy}}$