I am solving the following problem. How can I solve it? I tried to use stopping time... but I don't know how to use... Because I just know definition of stopping time and don't have any sense to apply this concept. Please help me. Let $Y_{1}, Y_{2}, \ldots$ be independent and equal $\pm 1$ with probability $\frac{1}{2} .$ Let $r, s$ be positive integers and define a stopping time $\boldsymbol{n}^{*}=\left(\text { first } n \text { such that } S_{n}=r \text { or }-s\right)$ where $S_n$ is sum of $Y_1 , \cdots , Y_n$.
Show
(a) $E \boldsymbol{n}^{*}<\infty$
(b) $P\left(S_{n^{*}}=r\right)=\frac{s}{r+s}$
For $r=s,$ evaluate $E e^{-\lambda n^{*}}, \lambda>0$
Some hints:
(a): if you haven't stopped yet, it is always possible to reach the stopping time within the next $r+s$ steps, so you can bound $n^*$ by a geometric random variable with finite mean.
(b): once you know (a) the optional stopping theorem applies, and the expected value of $S_n$ at the stopping time is $S_0$.