Difference between independence of events and independence of random variables

Question

I was wondering if someone could shed some light on the difference between independence of random variables and independence of events. The definitions look similar to me, but I am wondering if there are any conceptual differences between the two?

Answer 1

The relation is this: let $X,Y:\Omega \to \mathbb{R}$ two random variables. Then $X$ and $Y$ are independent if and only if their pre-image $\sigma $-algebras are independent.

In short: every two events of the form $X^{-1}(A)$ and $Y^{-1}(B)$, for arbitrary measurable $A,B\subset \mathbb{R}$, are independent.

Answer 2

Events $A_1, \ldots, A_n$ are mutually independent if $$P(\bigcap_{i \in S} A_i) = \prod_{i \in S} P(A_i)$$ for every subset $S \subseteq \{1,\ldots,n\}$. (In the case $n=2$, this is simply $P(A_1 \cap A_2) = P(A_1) P(A_2)$.)

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The definition of independence of random variables relies on the definition of independence of events, which is probably why the definitions look similar to you. Random variables $X_1, \ldots, X_n$ are mutually independent if the events $$\{X_1 \in E_1\}, \ldots, \{X_n \in E_n\}$$ are mutually independent for any measurable subsets $E_1, \ldots, E_n \subseteq \mathbb{R}$. (Equivalently, it suffices to check that $\{X_1 \le x_1\},\ldots,\{X_n \le x_n\}$ are mutually independent events for any real numbers $x_1, \ldots, x_n$.)