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Summary: Random walk, gambler's ruin and other good stuff.
Suppose 0 < p < 1 and let q = 1 - p. Let
X1, X2, . . . , Xn, . . .
be a sequence of mutually independent random variables with values in {-1, 1} such that
P(Xn = 1) = p, i = 1, 2, . . . , n, . . . .
It is not mathematically obvious that such a sequence of random variables exists; in fact, the sample space
must be uncountable. It is, however, intuitively clear that such a sequence, or something like it, exists.
For each n = 1, 2, . . . let
Sn =
n
i=1
Xi.
The sequence Sn, n = 1, 2, . . . is an example of what is called a simple random walk and is extraordinarily
useful.
Consider a game in which you win or lose a dollar at the n-th play if Xn is 1 or -1, respectively. You
start playing with a fortune of f dollars with the goal of winning w dollars. You quit if either you are broke
or you have won w dollars. What is the probability you will win? Go broke? How long will you play? We
now proceed to answer all these questions.
Let
T
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