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, $f_{X_1,X_2,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)=P[X_1=x_1,X_2=x_2,\mathrm{\dots },X_n=x_n]$.