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class5crofton

class5crofton - Urban Operations Research Professor Arnold...

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Unformatted text preview: Urban Operations Research Professor Arnold I. Barnett Lecture Note of 9/24/2001 Prepared by James S. Kang Crofton’s Method Let X 1 and X 2 be independent random variables that are uniformly distributed over the interval [0 ,a ]. We are interested in computing E [ | X 1 − X 2 | ]. For instance, in an urban setting, X 1 and X 2 may denote the location of an accident and the location where an emergency vehicle is currently parked in a road segment of length a , respectively. In this case, we want to know the distance (or the travel time) on average between the two locations, E [ | X 1 − X 2 | ]. We may solve this question using a joint distribution of X 1 and X 2 , but Crofton’s method is quite useful for the question. Let G ( a ) ≡ E [ | X 1 − X 2 | ]. Now consider the following question: If the interval were [0 ,a + ε ] where ε is smal l, what would G ( a + ε ) be? Table 1 summarizes G ( a + ε ) depending on the locations of X 1 and X 2 ....
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class5crofton - Urban Operations Research Professor Arnold...

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