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Joint Distribution Function

A joint distribution function is a Distribution Function $D(x,y)$ 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
D[(x,y)\in C)] = \int\!\!\!\int _{(X,Y)\in C} P(X,Y)\,dX\,dY
\end{displaymath} (4)

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)

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

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


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

© 1996-9 Eric W. Weisstein