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Using the preceding two problems prove the Theorem The number of visits to the point a > 0 that take place prior to the first return to the origin has expectation (p/q)a when p < q and 1 when p = q.
The truly amazing implications of the theorem are best explained in the language of fair games. A perfect coin is tossed until the first equalization of the accumulated numbers of heads and tails. The gambler receives one penny for every time the accumulated number of heads exceeds the accumulated number of tails by m. The ”fair entrance fee” for playing this game equals 1, independently of m!.
This problem is adapted from W. Feller, in Introduction to Probability Theory and Applications, Volume I, Chapter XIV, Section 9, problem 5, page 367.
The number of visits to a without first returning to the origin is a geometric random variable with ratio (2 * a - 1)/(2 * a). Therefore the expected value is
Using a standard series summation formula (easily derived by differentiating both sides of the geometric sum formula
with respect to r) namely
we see that the desired sum is
The interpretation in terms of the fair game is seen by associating a heads with a +1 and a tails with -1 in the random walk, or equivalently the sum of independent, identically distributed, Bernoulli random variables. Then the accumulated number of heads over the number of tails after n throws of the coin is just
or equivalently the position in the random walk. This is astonishing since the entrance fee is the same whether m = 5, meaning that a ”penny” is earned every time the number of heads exceeds the number of tails by 5 before the number of heads and tails is equal, ending the game, (not unreasonable in our experience) or if m = 1, 000, 000, which does seem unreasonable by comparison!
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