joint discrete density function
JointDiscreteDensityFunction
2001-10-26 19:32:52
2004-03-03 04:40:34
Definition
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Let $X_1, X_2, ..., X_n$ be $n$ random variables all defined on the same probability space. The \textbf{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:\\
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$f_{X_1, X_2, ..., X_n}: R^n \to R$\\
$f_{X_1, X_2, ..., X_n}(x_1,x_2,...,x_n) = P[X_1 = x_1, X_2 = x_2, ... , X_n = x_n]$\\
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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.\\
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Also, as in the case where $n=1$, this function satisfies:\\
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\begin{enumerate}
\item $f_{X_1, X_2, ..., X_n}(x_1,...,x_n) \geq 0$ $\forall (x_1,...,x_n)$
\item $\sum_{x_1, ... ,x_n}^{} { f_{X_1, X_2, ..., X_n}(x_1,...,x_n) }= 1$
\end{enumerate}
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In this case, $f_{X_1, X_2, ..., X_n}(x_1,...,x_n) = P[ X_1 = x_1, X_2 = x_2, ... , X_n = x_n ]$.