joint discrete density function
Let ${X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}$ be $n$ random variables^{} all defined on the same probability space^{}. The joint discrete 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 following function:
${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}:{R}^{n}\to R$
${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})=P[{X}_{1}={x}_{1},{X}_{2}={x}_{2},\mathrm{\dots},{X}_{n}={x}_{n}]$
As in the single variable case, sometimes it’s expressed as ${p}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ 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.
${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({x}_{1},\mathrm{\dots},{x}_{n})\ge 0$ $\forall ({x}_{1},\mathrm{\dots},{x}_{n})$

2.
${\sum}_{{x}_{1},\mathrm{\dots},{x}_{n}}{f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({x}_{1},\mathrm{\dots},{x}_{n})=1$
In this case, ${f}_{{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}}({x}_{1},\mathrm{\dots},{x}_{n})=P[{X}_{1}={x}_{1},{X}_{2}={x}_{2},\mathrm{\dots},{X}_{n}={x}_{n}]$.
Title  joint discrete density function 

Canonical name  JointDiscreteDensityFunction 
Date of creation  20130322 11:54:55 
Last modified on  20130322 11:54:55 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  10 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 60E05 
Synonym  joint probability function 
Synonym  joint distribution 