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Let $S_n=X_1+X_2+...+X_n$, where $X_i=1$ with probability $p$ and $X_i=-1$ with probability $q=1-p$, for all $i$ and independently of each other. Assume that $S_0=0$ and $0.

Show $$E\left(\sup\limits_{0\le k\le n}S_k\right) \le \frac{p}{q-p}$$

I would like to know how to prove it.

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    Is $q=1-p$? If not, what are the other possibilities for $X_i$?2012-08-06
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    @RossMillikan Yes, q=1-p2012-08-06

2 Answers 2