Lesson 6

Let \(\Pr[\alpha=1]=p\), \(\Pr[\alpha=0]=1-p\). As we already know, \(\operatorname{E}[\alpha]=\Pr[\alpha=1]=p\). An indicator variable is convenient because \(\alpha^{2}=\alpha\), hence \(\operatorname{E}[\alpha^{2}]=p\). Therefore, \[\operatorname{Var}[\alpha]=\operatorname{E}[\alpha^{2}]-\operatorname{E}[\alpha]^{2}=p-p^{2}=p(1-p) \ .\]