Expected value stopping time $E(\tau)$ with $\tau=\inf\{k>0 \mid X_k=1\} $

Question

I am a little confused about the expectation $E(\tau)$ of the stopping time $$\tau=\inf\{k>0 \mid X_k=1\}, $$with the $X_i$ independent random variables with values $-1$ and $1$, where $P(X_1=1)=1/2$ and $P(X_1=-1)=1/2$. I thought this is $$E(\tau)=1\cdot P(\tau=1)+2\cdot P(\tau=2)+\dots=\sum_{i=1}^{\infty}\frac{n}{2^n}.$$ Is this correct and what is the value of the series? Thanks for the help

Answer 1

Yes, looks good (just as a side remark: it should read $\sum_{i=1}^{\infty}$ instead of $\sum_{n=1}^{\infty}$). In order to calculate $\sum_{i=1}^{\infty} \frac{i}{2^i}$ use that

$$\sum_{i=1}^{\infty} i x^i = \sum_{i=1}^{\infty}(i+1) x^i - \sum_{i=1}^{\infty} x^i = \frac{d}{dx} \left( \sum_{i=1}^{\infty} x^{i+1} \right) - \sum_{i=1}^{\infty} x^i$$

for any $x \in (-1,1)$.