Joint Distribution Function

A joint distribution function is a distribution function D(x,y) in two variables defined by

D(x,y) = P(X<=x,Y<=y)

(1)

D_x(x) = lim_(y->infty)D(x,y)

(2)

D_y(y) = lim_(x->infty)D(x,y)

(3)

so that the joint probability function satisfies

(4)

(5)

D(x,y) = P{X in (-infty,x],Y in (-infty,y]}

(6)

(7)

(8)

Two random variables X and Y are independent iff

D(x,y)=D_x(x)D_y(y)

(9)

for all x and y and

(10)

A multiple distribution function is of the form

D(x_1,...,x_n)=P(X_1<=x_1,...,X_n<=x_n).

(11)

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More things to try:

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