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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 3437 Noncredit Exercises: Chapter 6: Problems 26, 2830, 51, 54 Theoretical Exercises 8, 14 ,22, 23, 33,; SelfTest 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 ≤ α X } where 0 < α ≤ 1? (d) What is P { Y ≤ α X } where 1 < α < ∞ ? (e) If Z = Y / X , find P { Z ≤ α } for all α, < α < ∞ . (f) Find the pdf f Z ( α ). Be sure to specify the value of f Z ( α ) for all α, ∞ < α < ∞ . 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 λ 1 and λ 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 λ 1 = λ 2 = λ ....
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 Spring '10
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 Standard Deviation, Variance, Probability theory, probability density function

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