Homework 6 Solution

Homework 6 Solution - EE 351K PROBABILITY& RANDOM...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

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.

Page1 / 5

Homework 6 Solution - EE 351K PROBABILITY& RANDOM...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online