01q1sol - Urban Operations Research Compiled by James S....

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Unformatted text preview: Urban Operations Research Compiled by James S. Kang Fall 2001 Quiz 1 Solutions 10/29/2001 Problem 1 (Kang, 2001) Let X 1 and X 2 be independent random variables denoting the two picks that are uniformly dis- tributed over the interval [0 ,a ]. Let G ( a ) E [ X 2 ] E [(max( X 1 ,X 2 )) 2 ]. Suppose a < X 1 a + and X 2 a . G ( a + ) for this case is computed as follows: a + a + G ( a + ) = E [(max( X 1 ,X 2 )) 2 ] = E [ X 1 2 ] = x 2 1 f X 1 ( x 1 ) dx 1 = 1 x 1 2 dx 1 a a a + 1 1 3 2 = x 1 = a + a + o ( ) , 3 a where o ( ) represents higher order terms of satisfying lim o ( ) = 0. Ignoring o ( ), we have the following table that summarizes G ( a + )s. Case X 1 a , 0 X 2 a a < X 1 a + , 0 X 2 a X 1 a , a < X 2 a + a < X 1 a + , a < X 2 a + Probability of a case a a = ( a ) 2 a + a + a + a a = a + a + ( a + ) 2 a a = a + a + ( a + ) 2 = ( ) 2 a + a + a + G ( a + ) given a case G ( a ) a 2 + a a 2 + a We do not care. a Using the total expectation theorem, we obtain 2 a 2 a 2 G ( a + ) = G ( a ) + ( a + a ) + ( a + a ) + o ( 2 ) a + ( a + ) 2 ( a + ) 2 2 a 2 a G ( a ) + 2( a + a ) a + ( a + ) 2 From the formula of the sum of an infinite geometric series, we know a 1 2 3 = = 1 + + a + 1 + a a a a Ignoring higher order terms of , we have a 1 a +...
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01q1sol - Urban Operations Research Compiled by James S....

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