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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:
- $f_{X_1, X_2, ..., X_n}(x_1,...,x_n) \geq 0$ $\forall (x_1,...,x_n)$
- $\sum_{x_1, ... ,x_n}^{} { f_{X_1, X_2, ..., X_n}(x_1,...,x_n) }= 1$
In this case, $f_{X_1, X_2, ..., X_n}(x_1,...,x_n) = P[ X_1 = x_1, X_2 = x_2, ... , X_n = x_n ]$
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