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Random Variable Domain Variable Representable Functor Function Measure

1	Representable Functor
1	Random Variable
1	Measure
1	Variable
1	Function
1	Domain

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Joint Continous Density Function

Let be random variables all defined on the same probability space. The joint continuous density function of , 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

$\displaystyle \int_D { f_{X_1,X_2,..., X_n}(u_1,u_2,...,u_n) du_1 du_2 ... du_n } =$ Prob $\displaystyle ({X_1,X_2,...,X_n}\in D)$

As in the case where , 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.