Let $p \in [0,1]$ and $\{X_i\}_{i =1}^\infty$ be a sequence of $\{-1, 1\}$-valued i.i.d random variables taking value 1 with probability $p$. We can consider the biased random walk $S_t$ defined at time $t$ by $$ S_t = \sum_{i=1}^t X_t. $$ Let $d > 0$ and define a stopping time $\tau$ by $\tau = \min \{ t \in \mathbb{Z}^+ : S_t \in \{-d, d\} \}$. What is the expectation of $\tau$?
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possibly related https://math.stackexchange.com/questions/2400386/biased-random-walk-in-1d-expected-hitting-time-for-either-edge-of-box – prosinac Apr 02 '20 at 22:15
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You are right! I didn't see this question when I was looking before I posted. – Harry Crimmins Apr 02 '20 at 23:12
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what you've written is a pure birth process -- a stopping threshold of -d doesnt make any sense and it is misleading to call this a random walk. That said, I'd guess what you meant was that if $X_i$ are iid Bernouli's then $S_t = \sum_{i=1}^t (2X_t-1)$... alternatively you can use the formula you've written but instead define the $X_i$ to be biased Rademachers. . – user8675309 Apr 03 '20 at 00:54