Random Variables Joint Law

Why

We name the image measure of a collection of real-valued random variables.

Definition

The joint law of a sequence of $n$ real-valued random variables is the image measure of the tuple-valued function whose components are the individual random variables.

Notation

Let $(X, \mathcal{A} , \mu )$ be a probability space and $(Y, \mathcal{B} )$ be a measurable space. Let $f_1, \dots , f_n: X \to Y$ be random variables. Define $f: X \to Y^n$ by $(f(x))_i = f_i(x)$. The joint law is the image measure of $f$.

We denote the joint law of $\set{f_i}$ by $\rvlaw{\mu }{f_1, \dots , f_n}: \mathcal{A} \to \nneri$. We defined it by

\[ \rvlaw{\mu }{f_1, \dots , f_n}(A) = \mu (\Set*{x \in X}{f(x) \in A}). \]