Independence of random variables - Imported Knowlpedia Content

A family of random variables $(X_i)_{i\in I}$ on a probability space $(\Omega,\mathcal F,\mathbb P)$ is independent if for every finite choice of indices $i_1,\dots,i_k\in I$ and every choice of Borel sets $A_1,\dots,A_k\subseteq\mathbb R$ ,

\mathbb P\big(X_{i_1}\in A_1,\dots,X_{i_k}\in A_k\big)=\prod_{j=1}^k \mathbb P\big(X_{i_j}\in A_j\big).

Equivalent characterizations

Equivalently, the sigma-algebras $\sigma(X_i)$ generated by the variables are independent.

Remarks

This says that all events of the form $\{X_i\in A\}$ behave like independent events under probability.

Examples

Let $\Omega=\{0,1\}^2$ with $\mathbb P$ uniform, and define $X(\omega_1,\omega_2)=\omega_1$ , $Y(\omega_1,\omega_2)=\omega_2$ . Then $X$ and $Y$ are independent random variables.
Let $\Omega=[0,1]^2$ with the product Lebesgue measure (normalized to a probability measure), and set $X(u,v)=u$ , $Y(u,v)=v$ . Then $X$ and $Y$ are independent and each has the uniform distribution on $[0,1]$ .