Can anyone provide the first hitting time distribution for a discrete random walk?
Edit: Specifically, a 1D random walk, starting at $k=0$. Each step moves either $-1$ or $+1$ without any boundaries. I require the distribution for the first hitting time at some arbitrary point $m>0$.
I cannot find it anywhere. I can only find it for continuous Brownian motion.
Best Answer
For $n,m\geq 1$, $$P(\tau(m)=n)=\begin{cases} \displaystyle{{m\over n}\binom{n}{(n+m)/2}{1\over 2^n}} &\mbox{if } m+n \mbox{ is even}\\[5pt] 0 &\mbox{if }m+n \mbox{ is odd}.\end{cases} $$