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# joint continuous density function

Let $X_{1},X_{2},...,X_{n}$ be $n$ random variables all defined on the same probability space. The joint continuous density function of $X_{1},X_{2},...,X_{n}$, denoted by $f_{{X_{1},X_{2},...,X_{n}}}(x_{1},x_{2},...,x_{n})$, is the function $f_{{X_{1},X_{2},...,X_{n}}}:\mathbb{R}^{n}\to\mathbb{R}$ such that for any domain $D\subset\mathbb{R}^{n}$, we have

$\displaystyle\int_{D}{f_{{X_{1},X_{2},...,X_{n}}}(u_{1},u_{2},...,u_{n})du_{1}% du_{2}...du_{n}}=\text{Prob}({X_{1},X_{2},...,X_{n}}\in D)$ |

As in the case where $n=1$, this function satisfies:

1. $f_{{X_{1},X_{2},...,X_{n}}}(x_{1},...,x_{n})\geq 0$ $\forall(x_{1},...,x_{n})$

2. $\int_{{x_{1},...,x_{n}}}{f_{{X_{1},X_{2},...,X_{n}}}(u_{1},u_{2},...,u_{n})du_% {1}du_{2}...du_{n}}=1$

Keywords:

statistics

Synonym:

joint mass function, joint density function, joint distribution

Type of Math Object:

Definition

Major Section:

Reference

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