APPENDIX B SOLUTIONS TO PROBLEMSB.1Before the student takes the SAT exam, we do not know – nor can we predict with certainty – what the score will be. The actual score depends on numerous factors, many of which we cannot even list, let alone know ahead of time. (The student’s innate ability, how the student feels on exam day, and which particular questions were asked, are just a few.) The eventual SAT score clearly satisfies the requirements of a random variable. B.3(i) Let Yitbe the binary variable equal to one if fund ioutperforms the market in year t. By assumption, P(Yit= 1) = .5 (a 50-50 chance of outperforming the market for each fund in each year). Now, for any fund, we are also assuming that performance relative to the market is independent across years. But then the probability that fund ioutperforms the market in all 10 years, P(Yi1= 1,Yi2= 1, …, Yi,10= 1), is just the product of the probabilities: P(Yi1= 1)⋅P(Yi2= 1) …P(Yi,10= 1) = (.5)10= 1/1024 (which is slightly less than .001).
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