# 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. 1.

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

2. 2.

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

As in the single variable case, $f_{X_{1},X_{2},...,X_{n}}$ does not represent the probability that each of the random variables takes on each of the values.

Title joint continuous density function JointContinuousDensityFunction 2013-03-22 11:54:58 2013-03-22 11:54:58 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 60A10 joint mass function joint density function joint distribution