part31

# 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 fx , y,z f z,y/x f x XYZ ZY/X X bg b g b g = fz , y/x f z/x,y f y ZY/X Z/XY Y b g = () ( ) ( ) / || XYZ Y X X Z/XY , y , z f z x , y fy x f x =

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2 Problem 2 4.48 Since X and Y are exponential random variables, then X0 and Y0 . Now we start with Fz PZ z Z bg Fz PXY z Z =- £ 0 −≥ −≤ ≥− when X Y X Y z X Y z Yxz 0 ≤+ when X Y X Y z Y X z y y xz =+ y z z x
3 Problem 2 – 4.48 – Continued: Therefore, XY z xz Y x z = - ££ + mr lq f= + + - + z zz z F z fx , y d y d x , y d y d x ZX Y 0 xz 0 z XY z bg b g b g using () y x Y X XY x y , y e e −λ λ in the above expression for F z Z , and integrating with respect to y and x: - + - + - - Fz 1 e e Z x xy y z y x z l ll l l l using fz d dz Fz ZZ = e e Z y z x z ej = + + - - l l

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4 Problem 3 4.51 In this case, both X and Y only take on non-negative values: 0 X and 0 Y .
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## This note was uploaded on 11/29/2009 for the course EE 131A taught by Professor Lorenzelli during the Fall '08 term at UCLA.

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

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