PS13 - University of Illinois Fall 2009 ECE 313: Problem...

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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Problem Set 13 Functions of Random Variables Due: Friday December 4 at 4 p.m. Reading: Ross, Chapter 6 except Sections 6.6 and 6.8; Powerpoint Lecture Slides, Sets 34-37 Noncredit Exercises: Chapter 6: Problems 26, 28-30, 51, 54 Theoretical Exercises 8, 14 ,22, 23, 33,; Self-Test Problems 3, 5, 6, 7, 12 1. [A piece of cake? or a sheet cake with a piece missing?] The jointly continuous random variables X and Y have joint pdf f X , Y ( u,v ) = 4 3 , < u < 1 , < v < 1 , max { u,v } > 1 2 , , elsewhere. (a) Sketch the u- v plane and indicate on it the region where f X , Y ( u,v ) is nonzero. (b) Find the marginal pdf f Y ( v ) of Y . (c) What is P { Y X } where 0 < 1? (d) What is P { Y X } where 1 < < ? (e) If Z = Y / X , find P { Z } for all , < < . (f) Find the pdf f Z ( ). Be sure to specify the value of f Z ( ) for all ,- < < . 2. [One function of two random variables] The joint pdf of X and Y is given by f X , Y ( u,v ) = ( 2 u, < u < 1 , < v < 1 , , elsewhere. Find the pdf of Z = X 2 Y . 3. [Independent exponential random variables] Let X 1 and X 2 be independent exponential random variables with parameters 1 and 2 respectively. (a) Find the pdf of Y = X 1 + X 2 by convolving the pdfs of X 1 and X 2 for the case 1 = 2 = ....
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PS13 - University of Illinois Fall 2009 ECE 313: Problem...

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