PlanetMath: joint continuous density function

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joint continuous density function

(Definition)

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.

"joint continuous density function" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Other names:

joint mass function, joint density function, joint distribution

Keywords:

statistics

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Cross-references: represent, variable, domain, function, probability space, random variables
There are 6 references to this entry.

This is version 6 of joint continuous density function, born on 2001-10-26, modified 2006-10-25.
Object id is 576, canonical name is JointContinousDensityFunction.
Accessed 24218 times total.

Classification:

AMS MSC:

60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)

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