05quiz1 - Logistical and Transportation Planning Methods

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Unformatted text preview: Logistical and Transportation Planning Methods 1.203J/6.281J/13.665J/15.073J/16.76J/ESD.216J Massachusetts Institute of Technology Cambridge, Massachusetts Quiz #1 October 26, 2005 OPEN BOOK TWO HOURS 5 PAGES, 3 PROBLEMS PLEASE SHOW ALL YOUR WORK! Inspirational quote: You've got to be very careful if you don't know where you are going because you might not get there. Yogi Berra, once a baseball catcher for the New York Yankees 35 points. Problem 1. Regions of Inner Coverage Consider two random points, ( X 1 , Y 1 ) and ( X 2 , Y 2 ), located uniformly and independently within a unit area square. These points are the locations of two independently patrolling police cars. Travel distance D 12 between the two points is the usual Manhattan or right- angle or L 1 distance metric, D 12 = | X 1- X 2 | + | Y 1 - Y 2 | The inner region of coverage formed by these two police cars at any time is the set of points in the rectangle formed by their locations, as shown in the figure. inner region of coverage ( X 1 , Y 1 ) ( X 2 , Y 2 ) 1 1 We are interested in the area of the inner region of coverage . (a) 10 pts. Find the mean area of the inner region of coverage . (b) 10 pts. Find the variance of the area of the inner region of coverage . (c) 15 pts. We are interested in the pdf of the area of the inner region of coverage . In our four-step process to obtain this, carefully work through steps 1, 2 and 3, and set up carefully and precisely but (in Step 4) do not compute any integrals you develop....
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05quiz1 - Logistical and Transportation Planning Methods

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