Lesson 4

Before providing several illustrative examples, let's state a generalization of the theorem.

Theorem (Linearity of Expectation). For any random variables \(\alpha_{1}, \dotsc, \alpha_{n} \colon U \to \mathbb{R}\) with finite expectations and any constants \(c_{1}, \dotsc, c_{n} \in \mathbb{R}\) the following holds: \[\operatorname{E}\left[\sum_{i \in [n]}c_{i} \alpha_{i}\right]=\sum_{i \in [n]}c_{i}\operatorname{E}[\alpha_{i}]  .\]