JointContinousDensityFunction 2001-10-26 19:44:36 2006-10-25 00:15:55 Definition statistics \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $X_1, X_2, ..., X_n$ be $n$ random variables all defined on the same probability space. The \textbf{joint continuous 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 function $f_{X_1, X_2, ..., X_n}: \mathbb{R}^n \to \mathbb{R}$ such that for any domain $D\subset \mathbb{R}^n$, we have \begin{align*} \int_D { f_{X_1,X_2,..., X_n}(u_1,u_2,...,u_n) du_1 du_2 ... du_n } = \text{Prob}({X_1,X_2,...,X_n}\in D) \end{align*} \par As in the case where $n=1$, this function satisfies:\\ \par \begin{enumerate} \item $f_{X_1, X_2, ..., X_n}(x_1,...,x_n) \geq 0$ $\forall (x_1,...,x_n)$ \item $\int_{x_1, ... ,x_n}^{} { f_{X_1, X_2, ..., X_n}(u_1,u_2,...,u_n) du_1 du_2 ... du_n }= 1$ \end{enumerate} \par As in the single variable case, $f_{X_1, X_2, ..., X_n}$ does not represent the probability that each of the random variables takes on each of the values.