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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 Problem 1 Assume that the random variables x and y are jointly normal random variables with density function given by: 2 2 2 2 2 2 1 1 2 2 1 2 1 ( , ) exp ( ) x y x y x y x y rxy f x y r r σ σ σ σ πσ σ = + Where r =0.5, 1 X Y = = σ σ . Define the new random variables w and z as: z xcos ysin w xsin ycos = = + θ θ θ 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 σ σ = and define the new random variables R and θ such that: = x = Rcos y Rsin and θ θ Find the joint PDF of R and θ ; i.e. ( , ) f R θ . Are R and θ independent ? Use: ( ) 1 2 1 1 tan ( ) d x dx x = + d) If x z = y prove that: 1 2 1 1 2 1 ( ) tan y x z x z r F z r σ σ π σ = + prove that 1 (x 0,y 0) (x 0,y 0) 2 P P > > + < > = Problem 2
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  • Spring '10
  • TTTTT
  • Probability theory, joint PDF, conditional density, Yarmouk University Hijjawi Faculty of Engineering and Technology Department of Communication Engineering

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