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.
Random walk and its expectation
3
$\begingroup$
probability
random-walk
-
3Is $q=1-p$? If not, what are the other possibilities for $X_i$? – 2012-08-06
-
0@RossMillikan Yes, q=1-p – 2012-08-06