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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

D[(x,y) in C)]=intint_((X,Y) in C)P(X,Y)dXdY
(4)
D(x in A,y in B)=int_(Y in B)int_(X in A)P(X,Y)dXdY
(5)
D(x,y)=P{X in (-infty,x],Y in (-infty,y]}
(6)
=int_(-infty)^xint_(-infty)^yP(X,Y)dXdY
(7)
D(a<=x<=a+da,b<=y<=b+db) =int_b^(b+db)int_a^(a+da)P(X,Y)dXdY approx P(a,b)dadb.
(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

P(x,y)=(partial^2D(x,y))/(partialxpartialy).
(10)

A multiple distribution function is of the form

D(x_1,...,x_n)=P(X_1<=x_1,...,X_n<=x_n).
(11)

See also

Distribution Function

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References

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

Referenced on Wolfram|Alpha

Joint Distribution Function

Cite this as:

Weisstein, Eric W. "Joint Distribution Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JointDistributionFunction.html

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