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Let $X_1, X_2, ..., X_n$ be $n$ random variables all defined on the same probability space. The 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
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As in the case where $n=1$ , this function satisfies:
- $f_{X_1, X_2, ..., X_n}(x_1,...,x_n) \geq 0$ $\forall (x_1,...,x_n)$
- $\int_{x_1, ... ,x_n}^{} { f_{X_1, X_2, ..., X_n}(u_1,u_2,...,u_n) du_1 du_2 ... du_n }= 1$
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.
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