Lesson 4

The expectation of \(\alpha\) can be computed directly from the definition: \[\begin{align*}\operatorname{E}[\alpha]&=\sum_{k=1}^{\infty}k(1-p)^{k-1}p=\\&=p\sum_{k=1}^{\infty}k(1-p)^{k-1}=\\&=p\left(-\sum_{k=0}^{\infty}(1-p)^{k}\right)'=\\&=p\left(-\frac{1}{p} \right)'=\\&=p \cdot \frac{1}{p^2}=\\&=\frac{1}{p} \ .\end{align*}\]