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Unformatted text preview: EE 351K PROBABILITY & RANDOM PROCESSES FALL 2011 Instructor: Sujay Sanghavi [email protected] Homework 6 Solution Problem 1 (a) A fire station is to be located at a point a along a road of length A , < A < ∞ . If fires will occur at points uniformly chosen on (0 ,A ) , where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to minimize the quantity E [ | X − a | ] when X is uniformly distributed over (0 ,A ) . (b) Now suppose that the road is of infinite length–stretching from point 0 outward to ∞ . If the distance of a fire from point 0 is exponentially distributed with rate λ , where should the fire station now be located? That is, we want to minimize E [ | X − a | ] with respect to a when X is now an exponential random variable with parameter λ . Sol : (a) We have that f X ( x ) = 1 /A , for x ∈ [0 ,A ] , and 0 otherwise. Thus E [ | X − a | ] = ∫ a ( a − x ) 1 A dx + ∫ A a ( x − a ) 1 A dx = 1 A ( A 2 2 − aA + a 2 ) = 1 A ( ( a − A 2 ) 2 + A 2 4 ) . Minimizing in terms of a , we will have that a − A 2 = 0 ⇒ a = A 2 . (b) We now have that f X ( x ) = λe- λx , for x ≥ , so that E [ | X − a | ] = ∫ a ( a − x ) λe- λx dx + ∫ A a ( x − a ) λe- λx dx = a + 1 λ 2 e- aλ + 1 λ . To minimize in terms of a , we compute d da E [ | X − a | ] = 1 − 2 e- aλ = 0 ⇒ a = ln 2 λ ....
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This note was uploaded on 02/20/2012 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas.
- Spring '07