joint continuous density function
Let ${X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}$ be $n$ random variables^{} all defined on the same probability space^{}. The joint continuous density 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 function ${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}:{\mathbb{R}}^{n}\to \mathbb{R}$ such that for any domain $D\subset {\mathbb{R}}^{n}$, we have
${\int}_{D}}{f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n})d{u}_{1}d{u}_{2}\mathrm{\dots}d{u}_{n}=\text{Prob}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}\in D)$ 
As in the case where $n=1$, this function satisfies:

1.
${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({x}_{1},\mathrm{\dots},{x}_{n})\ge 0$ $\forall ({x}_{1},\mathrm{\dots},{x}_{n})$

2.
${\int}_{{x}_{1},\mathrm{\dots},{x}_{n}}{f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n})\mathit{d}{u}_{1}\mathit{d}{u}_{2}\mathrm{\dots}\mathit{d}{u}_{n}=1$
As in the single variable case, ${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}$ does not represent the probability that each of the random variables takes on each of the values.
Title  joint continuous density function 

Canonical name  JointContinuousDensityFunction 
Date of creation  20130322 11:54:58 
Last modified on  20130322 11:54:58 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  11 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 60A10 
Synonym  joint mass function 
Synonym  joint density function 
Synonym  joint distribution 