Lesson 10

If \(\Pr[A]>0\), the conditional expectation of a random variable \(\alpha\) given event \(A\) is defined as \[\operatorname{E}[\alpha \mid A]=\sum_{a \in \mathbb{R}}a \cdot \Pr[\alpha =a \mid A] \ .\] For example, the expectation of the number rolled on a die given that it is at least four is \[\begin{align*}\operatorname{E}[\alpha \mid \alpha \ge 4]&=\sum_{i=1}^{6}i \cdot \Pr[\alpha=i \mid \alpha \ge 4]=\\&=1\cdot 0 + 2\cdot 0 + 3 \cdot 0 + 4 \cdot \frac{1}{3} + 5 \cdot \frac{1}{3} + 6 \cdot \frac{1}{3}=\\&=5\ .\end{align*}\]