Lesson 9

This property easily generalizes to a larger number of random variables. It is easy to see that pairwise independence is sufficient.

Theorem (linearity of variance for pairwise independent random variables). For any pairwise independent random variables \(\alpha_{1}, \dotsc, \alpha_{n}\), the following holds: \[\operatorname{Var}\left[\sum_{i=1}^{n}\alpha_{i}\right] = \sum_{i=1}^{n} \operatorname{Var}[\alpha_{i}] \ .\]