Simple Random Walk: two questions I am having difficulty in finding right resource to review. I am preparing interview on probability. One particular topic that I struggle the most is Simple Random Walk. I just want to know the following:
1) finding the first $n$ for which $S_n$ reaches a defined threshold $\alpha$.
2) the probability that $S_n$ reaches $\alpha$ for any given value of $n$.
3) Expected number of steps to reach an end point. 
I am wondering where I can find examples specifically for these 3 types of question?
Thanks. 
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Maybe I can try to answer my questions.  Assume symmetric. Assume we start from 0. 
1) the probability of first n for which $S_n$ reaches 10 is P($S_{10} =10) = 1/2^{10}$
2) for given value of $n$, we want to know the probability that $S_n$ reaches $\alpha$. This is equivalent as if asking $\max(S_1,S_2,S_3,\dots, S_n) \geq \alpha$. And the maximum of this probability formula is given by here : http://www.randomservices.org/random/bernoulli/Walk.html
3) This is simply the gambler ruin's problem, and you can look up the expected time for a player to ruin, which is $\alpha / (\alpha + \beta)$
 A: One useful book could be Probability and Random Processes 3rd ed, by Grimmett and Stirzaker. Sections 3.9 and 3.10 have material on Simple random walks. For the three questions:
1) I think you can use the hitting time theorem, p.79.
2) Here, you could use theorem (10), p. 78.
3) Eq. (9) on p. 74 gives a formula for the mean number of steps $D_k$, starting from $k$ before hitting one of the absorbing barriers at $0$ and $N$, for $p=1/2$, it is $D_k = k(N-k)$.
A: 
  
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*You might want to scroll through Simple random walks by Sven Erick Alm. Look at the examples there and if they are considered to be useful, I would like to put the focus on the first reference of the paper,  which is called a goldmine if someone likes to get more insight into simple random walks.
I fully agree and recommend to read the roughly $30$ pages of Chapter III: Fluctuations in Coin Tossing and Random Walks of the classic An Introduction to Probability Theory and Its Applications, Vol. I by W. Feller.
  
*Nice, easy-to-read examples are presented in this MIT paper titled Random Walks.

A: It seems to me that your questions are related to one-dimensional simple random walk with absorbing barriers. A popular application is gamblers ruin problem.  
I hope the following resources will be helpful, for a quick start:


*

*Markov Processes for Stochastic Modeling By Oliver Ibe; Chapter-8 

*Elements of Random Walk and Diffusion Processes By Oliver Ibe

*Introduction to Probability Models, By Sheldon M. Ross

