Fall 2010 HW11 - University of Illinois Fall 2010 ECE 313:...

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University of Illinois Fall 2010 ECE 313: Problem Set 11: Solutions Joint pdfs and functions of two random variables 1. [Joint densities] (a) f X,Y ( u,v ) = ± A (1 - ( u - v )) , u v A (1 + ( u - v )) , u < v Z 1 0 Z u 0 A (1 - u + v ) dvdu + Z 1 0 Z v 0 A (1 + u - v ) dudv = 1 A = 3 / 2 (b) The support of f X is the interval [0 , 1] . For 0 u 1 , f X ( u ) = Z f ( ) dv = Z u 0 A (1 - u + v ) dv + Z 1 u A ((1 + u - v ) dv Therefore, f X ( u ) = ± - 3 u 2 2 + 3 u 2 + 3 4 0 < u < 1 0 else. Since f X,Y ( ) = f X,Y ( v,u ) , or in other words, ( X,Y ) has the same pdf as ( Y,X ) , the pdfs f Y and f X are the same. (c) By symmetry, P { X > Y } = 1 / 2 . (d) First, use the pdf of X to compute P { X 1 2 } = R 1 0 . 5 f X ( u ) du = 0 . 5 . Also, P { X + Y < 1 ,X > 1 / 2 } is the integral of the joint pdf over the shaded region: u 1 0.5 1 v So P { X + Y < 1 1 / 2 } = R 1 0 . 5 R 1 - u 0 3 2 (1 - u + v ) dvdu = 3 32 . Therefore, P ( X + Y < 1 | X > 1 / 2) = P { X + Y < 1 1 / 2 } P { X > 1 / 2 } = 3 / 16 . 2. [Functions of random variables] The variable Z takes values in the positive reals, and for a 0 , F Z ( a ) = P { Z a } is equal to the integral of the joint pdf over the shaded region: a v u a/2
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This note was uploaded on 02/09/2011 for the course ECE 313 taught by Professor Milenkovic,o during the Fall '08 term at University of Illinois, Urbana Champaign.

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Fall 2010 HW11 - University of Illinois Fall 2010 ECE 313:...

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