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shawnacusteroyspr8
shawnacusteroyspr8
23.12.2019 • 
Mathematics

In the ivy league conference of the ncaa soccer championship, columbia university plays seven games per season. in this problem, we model columbia’s season as seven experiments and choose s = {w,n}7 for the underlying sample space, where w stands for win and n stands for a non-win (loss or tie). for i = , we define xi as a bernoulli random variable on s, signifying whether or not the i-th match results in a win. in this problem we assume that are i.i.d. and that the probability of a win is p. (the assumption that p is the same for all games is obviously questionable, for instance because we ignore the opponent’s strength, home-field advantage, and much more.) (a) write down three events on s; you are completely free to choose the events you write down. also give the probability of each of your events. (b) write down three outcomes in s; you are completely free to choose the outcomes you write down. also give the value x1 assigns to each of your three outcomes. (c) a run of k wins (‘winning streak’) corresponds to at least k successive wins in a row at any time in the season. suppose s(n,k) is the probability of having a run of k wins over n games. argue that for n ≥ 0 and k ≤ n, we have

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