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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 + ε and ≤ X 2 ≤ a . G ( a + ε ) for this case is computed as follows: a + ε a + ε G ( a + ε ) = 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 ε a a a + ε 1 1 3 2 = x 1 = a + aε + o ( ε ) , ε 3 a where o ( ε ) represents higher order terms of ε satisfying lim ε → o ( ε ε ) = 0. Ignoring o ( ε ), we have the following table that summarizes G ( a + ε )’s. Case ≤ X 1 ≤ a , 0 ≤ X 2 ≤ a a < X 1 ≤ a + ε , 0 ≤ X 2 ≤ a ≤ X 1 ≤ a , a < X 2 ≤ a + ε a < X 1 ≤ a + ε , a < X 2 ≤ a + ε Probability of a case a · a = ( a ) 2 a + ε a + ε a + ε ε a εa · = a + ε a + ε ( a + ε ) 2 a ε εa · = a + ε a + ε ( a + ε ) 2 ε · ε = ( ε ) 2 a + ε a + ε a + ε G ( a + ε ) given a case G ( a ) a 2 + aε a 2 + aε We do not care. εa Using the total expectation theorem, we obtain 2 a 2 εa 2 G ( a + ε ) = G ( a ) + ( a + aε ) + ( a + aε ) + o ( ε 2 ) a + ε ( a + ε ) 2 ( a + ε ) 2 2 a 2 εa ≈ G ( a ) + 2( a + aε ) a + ε ( a + ε ) 2 From the formula of the sum of an infinite geometric series, we know a 1 ε ε 2 ε 3 = = 1 − + − + ··· a + ε 1 + ε a a a a Ignoring higher order terms of ε , we have a ε ≈ 1 − a + ε...
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This note was uploaded on 11/08/2011 for the course AERO 16.72 taught by Professor Hansman during the Fall '06 term at MIT.
 Fall '06
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