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