Lesson 2

The expected value can also be infinite. For example, for the random variable \(\alpha\) such that \(\Pr[\alpha=2^{k}]=2^{-k}\) for all \(k \in \mathbb{Z}_{\ge 1}\): \[\operatorname{E}[\alpha]=\sum_{k \in \mathbb{Z}_{\ge 1}}2^{k} \cdot 2^{-k}=\sum_{k \in \mathbb{Z}_{\ge 1}}1=\infty \ .\] At the same time, for random variables taking a finite number of different values, the expected value is finite.