Consider a classical symmetric random walk that starts at origin $x=0$ and lasts for $N$ steps. If $x=-1$ is reached, a walk is terminated. What is the expected value of this process?
I divided a walk into 2 parts.
- A path hits $x=-1$ at some step $n \le N$. Then, I can compute a part of a total expected value summing the probabilities with weight $-1$ as it was presented in https://math.stackexchange.com/a/182376
- However, I encounter a problem with a part giving rise from walking only in the nonnegative part of the axis. How should I proceed?