Category:Random Vectors

This category contains results about Random Vectors.
Definitions specific to this category can be found in Definitions/Random Vectors.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $n \in \N$.

Let $\struct {S_1, \Sigma_1}$, $\struct {S_2, \Sigma_2}$, $\ldots$, $\struct {S_n, \Sigma_n}$ be measurable spaces.

Let:

$\ds S = \prod_{i \mathop = 1}^n S_i$

For each integer $1 \le i \le n$, let $X_i$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S_i, \Sigma_i}$.

Define a function $\mathbf X : \Omega \to S$ by:

$\map {\mathbf X} \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_n} \omega}$

for each $\omega \in \Omega$.

We call $\mathbf X$ a random vector.