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MIT6_042JF10_assn12

MIT6_042JF10_assn12 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science November 23, 2010 Tom Leighton and Marten van Dijk Problem Set 12 Problem 1. [15 points] In this problem, we will (hopefully) be making tons of money! Use your knowledge of probability and statistics to keep from going broke! Suppose the stock market contains N types of stocks, which can be modelled by independent random variables. Suppose furthermore that the behavior of these stocks is modelled by a double-or-nothing coin flip. That is, stock S i has half probability of doubling its value and half probability of going to 0. The stocks all cost a dollar, and you have N dollars. Say you only keep these stocks for one time-step (that is, at the end of this timestep, all stocks would have doubled in value or gone to 0). (a) [3 pts] What is your expected amount of money if you spend all your money on one stock? Your variance? (b) [3 pts] Suppose instead you diversified your purchases and bought N shares of all dif- ferent stocks. What is your expected amount of money then? Your variance? (c) [3 pts] The money that you have invested came from your financially conservative mother. As a result, your goals are much aligned with hers. Given this, which investment strategy should you take? (d) [3 pts] Now instead say that you make money on rolls of dice. Specifically, you play a game where you roll a standard six-sided dice, and get paid an amount (in dollars) equal to the number that comes up. What is your expected payoff? What is the variance? (e) [3 pts] We change the rules of the game so that your payoff is the cube of the number that comes up. In that case, what is your expected payoff? What is its variance?
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