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# ps1 - 1.203J 6.281J 13.665J 15.073J 16.76J ESD.216J...

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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J Logistical and Transportation Planning Methods Problem Set #1 Issued: September 11, 2006 Due: September 20, 2006 Problem 1 Two-horse race Consider a long-distance, two-horse race between horses A and B. The finishing time of Horse A is denoted by random variable (r.v.) X, which is uniformly distributed between 3 and 8 minutes. Horse B’s finishing time, Y, is uniformly distributed in [x/4, 2x] given that X=x. In other words, the conditional pdf for Y given X=x is: 4 x f Y | X ( y | x ) = for y 2 x . 7 x 4 (a) Please give the general relationship between the compound pdf of two arbitrary r.v.’s U and V , the conditional pdf of U given V = v and the pdf of V . Then give the general relationship between the compound pdf of U and V and the marginal probability density function of Y. (b) Derive the pdf of Y. (Hint: there are 3 possible cases to consider.) (c) Find f X | Y ( x | y ) , the conditional pdf of X given Y. (d) Let S be the event “Horse A wins the race”. Is± P ( S | Y = y ) = P ( S ) , i.e. are the event S and r.v. Y independent? (e) What is the probability that the winner will win by less than 1 minute? (f) What is the probability that the winner’s time will be less than 6 minutes? Problem 2 Cell Phones You are buying a cell phone and must select a monthly calling plan. The possible monthly calling plans are numbered from 1 to N (N is finite!). Plan i costs D i dollars per month and offers M i free minutes of cell phone conversation per month with C i additional cost per minute for each minute or part of a minute over the D i + 1 D i 0 maximum M i . Here i = 1,2,. .. N 1 we have M i + 1 M i 0 . 0 C i + 1 C i Problem Set #1 1

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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J Logistical and Transportation Planning Methods For plan i , M i is called its “free time”. So, for instance, plan 1 is the least expensive to purchase but has the least free time and the highest cost per minute of conversation beyond free time. On the other extreme, plan
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