Lesson 8

Informally, we call two random variables independent if no information about the value of one of these variables helps in determining the value of the other. Formally, random variables \(\alpha\) and \(\beta\) are called independent if for any \(a,b \in \mathbb{R}\), the events \[[\alpha=a] \text{ and }[\beta=b]\] are independent.

As with independent events, the concept of independence of random variables generalizes as follows. Random variables \(\alpha_{1}, \dotsc, \alpha_{n}\) are called mutually independent if for any \(a_{1}, \dotsc, a_{n} \in \mathbb{R}\), the events \[[\alpha_{1}=a_{1}], \dotsc, [\alpha_{n}=a_{n}]\] are independent. They are called \(k\)-independent if any \(k\) of them are mutually independent.