Lesson 2

Theorem. For any two random variables \(\alpha, \beta \colon U \to \mathbb{R}\) with finite expectations, the following equality holds: \[\operatorname{E}[\alpha + \beta]=\operatorname{E}[\alpha]+\operatorname{E}[\beta]  .\]

Proof. \[\begin{align*}\operatorname{E}[\alpha+\beta]&=\sum_{u \in U}(\alpha+\beta)(u)\Pr[u]=\\&=\sum_{u \in U}(\alpha(u)+\beta(u))\Pr[u]=\\&=\sum_{u \in U}\alpha(u)\Pr[u]+\sum_{u \in U}\beta(u)\Pr[u]=\\&=\operatorname{E}[\alpha]+\operatorname{E}[\beta]  .\end{align*}\]◼