joint cumulative distribution function

Let $X_1,X_2,\mathrm{\dots },X_n$ be $n$ random variables all defined on the same probability space. The joint cumulative distribution function of $X_1,X_2,\mathrm{\dots },X_n$, denoted by $F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,x_2,\mathrm{\dots },x_n)$, is the following function:

$F_{X_1,X_2,\mathrm{\dots },X_n}:R^n\to R$
$F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,x_2,\mathrm{\dots },x_n)=P[X_1\le x_1,X_2\le x_2,\mathrm{\dots },X_n\le x_n]$

As in the unidimensional case, this function satisfies:

1. ${lim}_{(x_1,\mathrm{\dots },x_n)\to (-\mathrm{\infty },\mathrm{\dots },-\mathrm{\infty })}{F}_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)=0$ and ${lim}_{(x_1,\mathrm{\dots },x_n)\to (\mathrm{\infty },\mathrm{\dots },\mathrm{\infty })}{F}_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)=1$

2. $F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)$ is a monotone, nondecreasing function.

3. $F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)$ is continuous from the right in each variable.

The way to evaluate $F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)$ is the following:

$F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)={\int }_{-\mathrm{\infty }}^{x_1}{\int }_{-\mathrm{\infty }}^{x_2}\mathrm{\cdots }{\int }_{-\mathrm{\infty }}^{x_n}{f}_{X_1,X_2,\mathrm{\dots },X_n}(u_1,\mathrm{\dots },u_n)𝑑u_1𝑑u_2\mathrm{\cdots }𝑑u_n$

(if $F$ is continuous) or

$F_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)={\sum }_{i_1\le x_1,\mathrm{\dots },i_n\le x_n}{f}_{X_1,X_2,\mathrm{\dots },X_n}(i_1,\mathrm{\dots },i_n)$

(if $F$ is discrete),

where $f_{X_1,X_2,\mathrm{\dots },X_n}$ is the joint density function of $X_1,\mathrm{\dots },X_n$.

Title	joint cumulative distribution function
Canonical name	JointCumulativeDistributionFunction
Date of creation	2013-03-22 11:54:52
Last modified on	2013-03-22 11:54:52
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Definition
Classification	msc 60A10
Synonym	joint cumulative distribution