Joint Distribution Function

A joint distribution function is a Distribution Function in two variables defined by

$\displaystyle D(x,y)$	$\textstyle \equiv$	$\displaystyle P(X\leq x, Y\leq y)$	(1)
$\displaystyle D_x(x)$	$\textstyle \equiv$	$\displaystyle \lim_{y\to\infty} D(x,y)$	(2)
$\displaystyle D_y(y)$	$\textstyle \equiv$	$\displaystyle \lim_{x\to\infty} D(x,y)$	(3)

so that the joint probability function satisfies

$\begin{displaymath} D[(x,y)\in C)] = \int\!\!\!\int _{(X,Y)\in C} P(X,Y)\,dX\,dY \end{displaymath}$

(4)

$\begin{displaymath} D(x\in A, y\in B) = \int_{Y\in B}\int_{X\in A} P(X,Y)\,dX\,dY \end{displaymath}$

(5)

$\displaystyle D(x,y)$	$\textstyle =$	$\displaystyle P\{X\in(-\infty,x], Y\in(-\infty,y]\}$
	$\textstyle =$	$\displaystyle \int^x_{-\infty} \int^y_{-\infty} P(X,Y)\,dX\,dY$	(6)

$\begin{displaymath} D(a \leq x \leq a+da, b \leq y \leq b+db) = \int^{b+db}_b \int^{a+da}_a P(X,Y)\,dX\,dY \approx P(a,b)\,da\,db. \end{displaymath}$

(7)

A multiple distribution function is of the form

$\begin{displaymath} D(x_1, \ldots, x_n) \equiv P(X_1\leq x_1, \ldots, X_n\leq x_n). \end{displaymath}$

(8)

See also Distribution Function

References

Grimmett, G. and Stirzaker, D. Probability and Random Processes, 2nd ed. New York: Oxford University Press, 1992.