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02q1soln

# 02q1soln - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology 1.203J/6.281J/13.665J/15.073J/16.76J/ESD.216J Logistical and Transportation P lanning Methods Fall 2002 Quiz 1 Solutions 1 (a)(i) Without loss of generality we can pin down X 1 at any fixed point. X 2 is still uniformly distributed over the square. Assuming that the police car will always follow a shortest route to the emergency incident, the max possible distance between X 1 and X 2 is 2 km. The travel distance is thus uniformly distributed between 0 and 2 km. (a)(ii) Following similar logic, the max possible distance is now 4 km. The travel distance is thus uniformly distributed between 0 and 4 km. (b) Let’s number the links as shown in Figure 1. There is a 1 chance that the emergency 12 incident will be on any one of the 12 links. Thus if we can determine the conditional pdf for the travel distance from X 1 (conditioned to be uniformly distributed on link 7) to X 2 for each possible link for X 2 , we are done. All we do then is add the resulting conditional pdfs, multiplying each 1 by 12 , the probability of occurrence. Careful inspection of the problem reveals that with regard to computing the conditional travel distance pdf between X 1 and X 2 there are three sets of links 1 km 3 8 1 4 police car here 5 2 6 9 10 7 11 12 Figure 1: Link Numbering Set 1: 1, 3, 4, 5, 6, 8, 9, 10, 11 Page 1 of 7

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• Fall '06
• hansman
• Probability theory, Massachusetts Institute of Technology, Cumulative distribution function, Massachusetts Institute, busy period

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