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Unformatted text preview: ECE 330:541, Stochastic Signals and Systems Midterm 1 Exam Solutions, Fall 2002 1. a) You must show f XY 6 = f X f Y . Calculate f X and f Y by f X ( x ) = Z √ 1- x 2- √ 1- x 2 1 π dy = 2 π p 1- x 2 for- 1 ≤ x ≤ 1 . Similarly, f Y ( y ) = 2 π q 1- y 2 for- 1 ≤ y ≤ 1 . Observe that f XY 6 = f X f Y . b) For the polar coordinate transformation, X = R cos(Θ) and Y = R sin(Θ). Calculate the Jacobian of the transformation to get J = r (easiest way is to do derivatives of x and y with respect to r and θ ) , so f R Θ ( r,θ ) = f XY ( x,y ) | J | = r π for r ∈ [0 , 1] and θ ∈ [0 , 2 π ]. Now calculate the marginals f R ( r ) = Z 2 π r π dθ = 2 r f Θ ( θ ) = Z 1 r π dr = 1 2 π . Observe f R Θ = f R f Θ . c) The easy way to do this is E ( XY ) = E [ R 2 cos(Θ)sin(Θ)] = E R [ R 2 ] E Θ [cosΘsinΘ] = 0 (here we have used the independence from part b to separate the expectations). We have also used the fact that E (cosΘsinΘ) = 0 (this is the integral of a periodic function over twice its period, hence 0). Also(cosΘsinΘ) = 0 (this is the integral of a periodic function over twice its period, hence 0)....
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This note was uploaded on 02/11/2012 for the course EEE 352 taught by Professor Ferry during the Spring '08 term at ASU.
- Spring '08