ecen303_sol9

ecen303_sol9 - ECEN 303 Random Signals and Systems Spring...

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ECEN 303: Random Signals and Systems Spring 2010 Solution to Homework #9 Problem 1: The joint PDF of X and Y is f ( x,y ) = 1 , 0 < x < 1 , 0 < y < 1 0 , otherwise 1. Find the PDFs of X and Y . Are X and Y independent? 2. Let Z = X + Y . Find the PDF and the expectation of Z . Solution. The PDF of X f X ( x ) = Z -∞ f ( x,y ) dy = 1 , 0 < x < 1 0 , otherwise Similarly, the PDF of Y f Y ( y ) = Z -∞ f ( x,y ) dx = 1 , 0 < y < 1 0 , otherwise We have f ( x,y ) = f X ( x ) f Y ( y ) , x,y R and hence X and Y are independent. The CDF of Z can be calculated as F Z ( z ) = Pr( Z z ) = Pr( X + Y z ) = Z Z x + y z f ( x,y ) dxdy = 1 2 z 2 , 0 z < 1 1 - 1 2 (2 - z ) 2 , z z < 2 0 , otherwise 1

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1 z x 0 z z 1 1 x 0 z z 1-z 1-z Case 1: 0<z<1 Case 2: 1<z<2 z 1 Hence the PDF of Z f Z ( z ) = d dz F Z ( z ) == z, 0 < z < 1 2 - z, 1 < z < 2 0 , otherwise The expectation of Z can be calculated as E [ Z ] = Z -∞ zf Z ( z ) dz = Z 1 0 z 2 dz + Z 2 1 z (2 - z ) dz = 1 3 + 2 3 = 1 Problem 2: Let X and Y be two independent, continuous random variables with PDF
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ecen303_sol9 - ECEN 303 Random Signals and Systems Spring...

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