# joint discrete density function

Let $X_{1},X_{2},...,X_{n}$ be $n$ random variables  all defined on the same probability space  . The joint discrete 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 following function:

$f_{X_{1},X_{2},...,X_{n}}:R^{n}\to R$
$f_{X_{1},X_{2},...,X_{n}}(x_{1},x_{2},...,x_{n})=P[X_{1}=x_{1},X_{2}=x_{2},...% ,X_{n}=x_{n}]$

As in the single variable case, sometimes it’s expressed as $p_{X_{1},X_{2},...,X_{n}}(x_{1},x_{2},...,x_{n})$ to mark the difference  between this function and the continuous joint density function.

Also, 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.

$\sum_{x_{1},...,x_{n}}{f_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})}=1$

In this case, $f_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})=P[X_{1}=x_{1},X_{2}=x_{2},...,X_{n}% =x_{n}]$.

Title joint discrete density function JointDiscreteDensityFunction 2013-03-22 11:54:55 2013-03-22 11:54:55 mathcam (2727) mathcam (2727) 10 mathcam (2727) Definition msc 60E05 joint probability function joint distribution