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# part31 - ECE 863 Analysis of Stochastic Systems Fall 2001...

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1 ECE 863 – Analysis of Stochastic Systems Fall 2001 Solutions for Homework Set #5 Michigan State University Department of Electrical & Computer Engineering Problem 1 4.41 f x,y,z f z,y/x f x XYZ ZY/X X b g b g b g = f z,y/x f z/x,y f y ZY/X Z/XY Y b g b g b g = ( ) ( ) ( ) ( ) / | | XYZ Y X X Z/XY f x,y,z f z x,y f y x f x =

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2 Problem 2 4.48 Since X and Y are exponential random variables, then X 0 and Y 0 . Now we start with F z P Z z Z b g = £ F z P X Y z Z b g = - £ 0 when X Y X Y z X Y z Y x z 0 + when X Y X Y z Y X z Y x z y y x z = + y x z = - z z x
3 Problem 2 – 4.48 – Continued: Therefore, X Y z x z Y x z - £ = - £ £ + m r l q fi = + + - + z z z z F z f x,y dydx f x,y dydx Z XY 0 x z 0 z XY x z x z z b g b g b g using ( ) ( ) ( ) y x Y X XY x y f x,y e e −λ −λ = λ λ in the above expression for F z Z b g , and integrating with respect to y and x: fi = - + - + - - F z 1 e e Z x x y y z y x y x z b g l l l l l l l l using f z d dz F z Z Z b g b g = f z e e Z x y x y y z x z b g e j = + + - - l l l l

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part31 - ECE 863 Analysis of Stochastic Systems Fall 2001...

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