In simple asymmetric random walk, starting at 0, with probability p, it moves to i+1, and with probability 1-p, it moves to i-1 when it at i. I want to calculate the variance of the time at which it reaches n, n>0.
When $n=1$, I can calculate the variance by using Z-transform for the equation $B_1 = 1*p + (1+B_{1} + B_{1})*(1-p)$, where $B_1$ means the time it takes from starting point 0 to 1. I got this equation by condition on the first step.
Actually, the distribution of $B_{i\to i+1}$ are independent and indetical for all $i$. So $Var(B_{0\to n}) = Var(\sum_{i=0}^{n-1}B_{i\to i+1}) = nVar(B_{0 \to 1})$.