Lesson 9

It is not difficult to show that the mathematical expectation of the random variable \(\operatorname{E}[\alpha \mid \beta]\) equals the mathematical expectation of the random variable \(\alpha\): \[\begin{align*}\operatorname{E}\left[\operatorname{E}[\alpha \mid \beta ]\right]&=\sum_{b}\operatorname{E}[\alpha \mid \beta=b] \Pr[\beta=b]=\\&=\sum_{b}\left(\sum_{a}a\Pr[\alpha=a \mid \beta=b]\right)\Pr[\beta=b]=\\&=\sum_{b}\left(\sum_{a}a\Pr[\alpha=a, \beta=b]\right)=\\&=\sum_{a,b}a\Pr[\alpha=a,\beta=b]=\\&=\operatorname{E}[\alpha] \ .\end{align*}\]