ps08_sol - University of Illinois Spring 2010 ECE 313...

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University of Illinois Spring 2010 ECE 313: Problem Set 8: Problems and Solutions CDFs; Continuous Random Variables Due: Wednesday March 17 at 4 p.m. Reading: Ross, chapter 5; Powerpoint Lecture Slides, Sets 20-22 1. [Maximization of the newsboy’s proﬁt] Each day, a newsboy buys newspapers from the publisher for c 1 cents each, sells them for c 2 cents each, and recycles the unsold papers (if any) getting c 3 cents for each. Note that c 2 > c 1 > c 3 . Let H denote the number of papers that the newsboy purchases each day. The demand for papers is a discrete random variable X that takes on nonnegative integer values. Do NOT assume that X is a binomial random variable. Let F ( u ) denote the CDF of X . (a) Express the probability that the newsboy is able to sell all H papers in terms of F ( u ). Solution: The newsboy sells all his papers if and only if the demand is H or more, i.e., if and only if the event { X H } occurs. P { X H } = 1 - P { X H - 1 } = 1 - F ( H - 1). (b) One day, the newsboy decides to buy one additional paper in the hopes of selling it and increasing his proﬁt. Express the probability that he is unable to sell the additional paper in terms of F ( u ). Be sure you understand the diﬀerence between “not being able to sell the ( H + 1)-th paper” and “being able to sell all H papers but not the extra ( H + 1)-th paper.” Solution: P { unable to sell extra paper } = P { demand is at most H } = P { X H } = F ( H ). (c) Find A ( H +1), the average additional proﬁt from the sale of the extra (that is, ( H +1)-th) paper. Solution: The newsboy is unable to sell the extra paper with probability F ( H ) and thus loses c 1 - c 3 cents with this probability , or is able to sell the extra paper with probability 1 - F ( H ) and thus makes c 2 - c 1 cents with this probability. Hence, his average additional proﬁt from the ( H + 1)-th paper is A

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ps08_sol - University of Illinois Spring 2010 ECE 313...

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