# To quickly derive the exist distribution it is

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Unformatted text preview: · +2k = 2k+1 1, this is true if we lose k times in a row before we win. Thus every time we win our net proﬁt is up by \$1 from the previous time we won. This system will succeed in making us rich as long as the probability of winning is positive, so where’s the catch? Suppose for simplicity we play 6 times. Let L be the time of the last win L (with L = 0 if there were six losses) and N be the total number of wins in the ﬁrst six plays. The number of the 64 outcomes that lead to the possible values of (L, N ) are 166 CHAPTER 5. MARTINGALES L 6 5 4 3 2 1 0 N =0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 5 4 3 2 1 3 10 6 3 1 4 10 4 1 5 5 1 6 1 If we lost six times in a row then our total losses are 63. If the last loss was at 1, 2, 3, 4, 5, or 6 then our losses after that win are 31, 15, 7, 3, 1 and 0. Taking into account that N wins means a total winning of \$N up to and including the last win we see that the distribution for positive values is (omitting the denominator which is always 64): 6 1 5...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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