Lesson 10

Let \(\delta_{k}=\alpha_{1}+ \dotsb+ \alpha_{k}\) be the sum of \(k\) independent identically distributed variables \(\alpha_{1}, \dotsc, \alpha_{k}\), and \(\gamma\) be an independent random variable taking values in \(\mathbb{Z}_{> 0}\). Then \[\begin{align*}\operatorname{E}[\delta_{\gamma}]&=\operatorname{E}[\operatorname{E}[\delta_{\gamma}\mid \gamma]]=\\&=\sum_{k \in \mathbb{Z}_{\ge 0}}\operatorname{E}[\delta_{\gamma} \mid \gamma=k] \cdot \Pr[\gamma=k]=\\&=\sum_{k \in \mathbb{Z}_{\ge 0}}(k\operatorname{E}[\alpha])\cdot \Pr[\gamma=k]=\\&=\operatorname{E}[\alpha] \cdot \sum_{k \in \mathbb{Z}_{\ge 0}}k\Pr[\gamma=k]=\\&=\operatorname{E}[\alpha] \cdot \operatorname{E}[\gamma] \ .\end{align*}\] This formula is known as Wald's identity.