hw5 - X and Y be given by f X,Y ( x,y ) = e-x y x...

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Virginia Tech The Bradley Department of Electrical and Computer Engineering ECE 5605 – Stochastic Signals and Systems – Fall 08 Problem Set 5 Problem 1. The random variables X and Y have joint density f X,Y ( x,y ) = ± e - x 0 x < , 0 y 1 0 otherwise Evaluate the probability P [ X Y ]. Problem 2. X and Y are independent and uniform in the interval (0 ,a ). Find the pdf of Z = X/Y . Problem 3. Let the random variables X and Y have joint density f X,Y ( x,y ) = ( x y ln 2 exp ² - x 2 2 ³ 0 x < , 1 y 2 0 otherwise Find the probability density of Z = X/Y . Problem 4. X and Y are independent uniform random variables in the common interval (0 , 1). Determine f Z ( z ), where Z = X + Y . Problem 5. X and Y are independent, random variables with common pdf f X ( x ) = e - x U ( x ) f Y ( y ) = e - y U ( y ) Find the pdf of the following random variables (a) X - Y , (b) XY , and (c) min( X,Y ). Problem 6. The joint pdf of the random variables X and Y is given by f XY ( x,y ) = ± 1 shaded area Fig. 1 0 otherwise Let Z = X + Y . Find F Z ( z ) and f Z ( z ). Problem 7. Let the joint pdf of
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Unformatted text preview: X and Y be given by f X,Y ( x,y ) = e-x y x &lt; otherwise 1 Figure 1: Problem 6. Dene Z = X + Y , W = X-Y . Find the joint pdf of Z and W . Show that Z is not an exponential random variable. Problem 8. Let f X,Y ( x,y ) = 2 e-( x + y ) &lt; x &lt; y &lt; otherwise Dene Z = X + Y , W = Y/X . Determine the joint pdf of Z and W . Problem 9. (a) Use the auxiliary variable method to nd the pdf of Z = X X + Y . (b) Find the pdf of Z if X and Y are independent exponential random variables with the same parameter . Problem 10. Suppose X and Y are zero-mean independent Gaussian random variables with common variance 2 . Dene R = X 2 + Y 2 and = tan-1 ( Y/X ), where | | &lt; . Obtain their joint density function. 2...
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hw5 - X and Y be given by f X,Y ( x,y ) = e-x y x...

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