HW3 - Yarmouk University Hijjawi Faculty of Engineering and...

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Yarmouk University Hijjawi Faculty of Engineering and Technology Department of Communication Engineering Random Processes CME 610 Homework Assignment 3 First Semester 2010/2011 Instructor: Dr. Khaled Gharaibeh Due Date 8/11/2010 P r o b l e m 1 Assume that the random variables x and y are jointly normal random variables with density function given by: 22 2 2 11 2 21 (, ) e x p () xy rxy fxy r r σσ πσ σ ⎛⎞ ⎜⎟ =− + ⎝⎠ ⎩⎭ Where r =0.5, 1 XY == σ . Define the new random variables w and z as: zx c o s y s i n wx s i n y c o s = =+ θ where θ = π /4. a) Find θ such that r ZW =0. b) Find the condition that makes the two random variables have circular symmetry. c) Let r =0 and x y = and define the new random variables R and θ such that: = x=Rcos y Rsin and Find the joint PDF of R and θ ; i.e. (,) fR . Are R and θ independent ? Use: ( ) 1 2 1 1 tan ( ) d x dx x = + d) If x z= y prove that: 1 2 2 1 t an yx z x zr Fz r π prove that 1 (x 0,y 0) 2 PP >> +<> = P r o b l e m 2 The joint PDF of x and y is given by 1 in the hashed area of the graph 0e l s e w h e r e = Find 02 , 03
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This note was uploaded on 12/29/2010 for the course UAE 605 taught by Professor Ttttt during the Spring '10 term at University of Arkansas – Fort Smith.

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HW3 - Yarmouk University Hijjawi Faculty of Engineering and...

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