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 $f_{X_1,X_2,\mathrm{\dots },X_n}:{ℝ}^n\to ℝ$ such that for any domain $D\subset {ℝ}^n$, we have

${\displaystyle {\int }_D}{f}_{X_1,X_2,\mathrm{\dots },X_n}(u_1,u_2,\mathrm{\dots },u_n)d{u}_{1}d{u}_2\mathrm{\dots }d{u}_n=\text{Prob}(X_1,X_2,\mathrm{\dots },X_n\in D)$

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. ${\int }_{x_1,\mathrm{\dots },x_n}{f}_{X_1,X_2,\mathrm{\dots },X_n}(u_1,u_2,\mathrm{\dots },u_n)𝑑u_1𝑑u_2\mathrm{\dots }𝑑u_n=1$

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	2013-03-22 11:54:58
Last modified on	2013-03-22 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