MidTerm_Exam_Fall2010_Soln_Irbid

MidTerm_Exam_Fall2010_Soln_Irbid - Yarmouk University...

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Yarmouk University Hijjawi Faculty of Engineering and Technology Department of Communication Engineering Random Processes CME 610 Midterm Exam Solution First Semester 2010/2011 Nov. 28, 2010 Instructor: Dr. Khaled Gharaibeh Time Allowed: 1 hour and 30 min P r o b l e m 1 6 p o i n t s You are given a normal random variable x ~ N ( 0 , 2). Define a new random variable y = a x + b . 1. Find the mean and variance of y . 2. Find the correlation of x and y (R xy ) 3. Find the correlation coefficient of x and y ( ρ xy ). 4. Find a and b such that x and y orthogonal? 22 2 2 2 2 2 2 2 a) 0 2 2 20 2 [y] [ x ] [x] ( ) [y ] [( x ) ] [x x ] [x ] [x ] ( ) )[ x y ] [x( x )] [ x x] [x ] [x] ( ) ( ) E[x]E ) y xy xy xy xy xy EE a b a E b ab b EEE a b b Ea b b a E abE b b aE a a bR E Ea b b bE a b a CR cr σ σσ =+ = + = + = =−=+ + =++ == = = = + = + = 2 2 2 2 2 1 2 For 0 you can simply set 0 for orthogonality, however, For 0 choose to have any value and then: 0 0 [y] ), x ] [x ] [x ] xy xy aa a a dR a RE aE bE aE b E = ⇒= += ⇒+ = =−

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Yarmouk University Hijjawi Faculty of Engineering and Technology Department of Communication Engineering P r o b l e m 2 6 p o i n t s The random variable x has a probability distribution function F x ( x
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MidTerm_Exam_Fall2010_Soln_Irbid - Yarmouk University...

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