Question
Suppose I have a particle at position $0$ that, every second, moves either 1 step right w.p. $p$ or 1 step left w.p. $1-p$. The particle stops when it reaches position $1$. If $T$ is the time it takes until the particle stops, what is $Var[T]$?
Thoughts
I know that $E[T]$ can be calculated by noting a simple linear relationship between when the particle is at $0$ vs when it is at $-1$; but I do not see how to approach finding the variance. Of course, I would like to be able to find an approach which easily generalizes (i.e. particle starts at $0$ and ends when it reaches $n>0$.
Also, this was in Ross Stochastic Processes chapter 1 (not homework, just piqued my interest) and so it should have an elementary solution.
As we discussed, you can note that $$ T = \left\{ \begin{array}{ll} 1 &\mbox{ , if first move right} \\ 1 + A + B & \mbox{ , if first move left} \end{array} \right.$$ where $A$ and $B$ are independent and have the same distribution as $T$. You can work out the details and perhaps create your own full answer to mark as "best answer" if you like.