Lesson 4

Expanding the brackets in the definition of variance, we obtain (after simplification) an expression that is often more convenient for calculations: \[\begin{align*}\operatorname{Var}[\alpha]&=\operatorname{E}[(\alpha - \operatorname{E}[\alpha])^{2}]=&\text{(expand brackets)}\\&=\operatorname{E}[\alpha^{2} - 2\alpha\operatorname{E}[\alpha]+\operatorname{E}[\alpha]^{2}]=&\text{(linearity)}\\&=\operatorname{E}[\alpha^{2}]-\operatorname{E}[2\alpha\operatorname{E}[\alpha]]+\operatorname{E}[\operatorname{E}[\alpha]^{2}]=\\&=\operatorname{E}[\alpha^{2}]-2\operatorname{E}[\alpha]\operatorname{E}[\alpha]+\operatorname{E}[\alpha]^{2}=\\&=\operatorname{E}[\alpha^{2}]-\operatorname{E}[\alpha]^{2} \ .\end{align*}\]