Editing Joint probability distribution (section)

===Continuous case===

The '''joint [[probability density function]]''' <math>f_{X,Y}(x,y)</math> for two [[continuous random variable]]s is defined as the derivative of the joint cumulative distribution function (see {{EquationNote|Eq.1}}):

{{Equation box 1
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|equation = {{NumBlk||<math>f_{X,Y}(x,y) = \frac{\partial^2 F_{X,Y}(x,y)}{\partial x \partial y}</math>|{{EquationRef|Eq.5}}}}
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This is equal to:
:<math>f_{X,Y}(x,y) = f_{Y\mid X}(y\mid x)f_X(x) = f_{X\mid Y}(x\mid y)f_Y(y)</math>

where <math>f_{Y\mid X}(y\mid x)</math> and <math>f_{X\mid Y}(x\mid y)</math> are the [[conditional distribution]]s of <math>Y</math> given <math>X=x</math> and of <math>X</math> given <math>Y=y</math> respectively, and <math>f_X(x)</math> and <math>f_Y(y)</math> are the [[marginal distribution]]s for <math>X</math> and <math>Y</math> respectively.

The definition extends naturally to more than two random variables:

{{Equation box 1
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|equation = {{NumBlk||<math>f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \frac{\partial^n F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)}{\partial x_1 \ldots \partial x_n}</math>|{{EquationRef|Eq.6}}}}
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Again, since these are probability distributions, one has
:<math>\int_x \int_y f_{X,Y}(x,y) \; dy \; dx= 1</math>
respectively
:<math>\int_{x_1} \ldots \int_{x_n} f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) \; dx_n \ldots \; dx_1 = 1</math>