Independent Random Variables

Why

What does it mean for two random variables to be independent? What are the events associated with a random variable?

Definition

Two random variables are independent if the sigma algebras generated by the random variables are independent. In general, a family of random variables are independent if the sigma algebras generated by the random variables are independent.

Notation

Let $(X, \mathcal{A} , \mu )$ be a probability space and $(Y, \mathcal{B} )$ be a measurable space. Let $f_1,f_2: X \to Y$ be random variables. If the random variables are independent we write $f_1 \perp f_2$.

Results

Let $f_1, \dots , f_n$ be independent real-valued random variables defined on a probability space $(X, \mathcal{A} , \mu )$.

Let $B_1, \dots , B_n$ be Borel sets of real numbers and let $A_i = f_i^{-1}(B_i)$. Let $A = \cap_{i = 1}^{n} f_i^{-1}(B_i)$. Then

\[ \mu (A) = \prod_{i = 1}^{n} \mu (A_i) \]

Since $f_i$ are independent, so are the sigma algebras they generate. $A_i$ are in each of these sigma algebras, so by definition of independence the measure of the intersection is the product of the measures.