Lesson 3

It is easy to see that when multiplying a random variable by a constant, its expected value is also multiplied by that constant: \[\operatorname{E}[c\alpha]=c\operatorname{E}[\alpha] \ .\] Indeed, all values of the random variable simply get multiplied by \(c\), meaning the mean also gets multiplied by \(c\). Formally: \[\begin{align*}\operatorname{E}[c\alpha]&=\sum_{u \in U}\Pr[u](c\alpha)(u)=\\&=\sum_{u \in U}\Pr[u]c\alpha(u)=\\&=c\sum_{u \in U}\Pr[u]\alpha(u)=\\&=c\operatorname{E}[\alpha] \ .\end{align*}\]