PS09 - University of Illinois Fall 2009 ECE 313: Problem...

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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Problem Set 9 Continuous Random Variables Due: Wednesday October 28 at 4 p.m. Reading: Ross, Chapter 5; Powerpoint Lecture Slides, Sets 22-24 Noncredit Exercises: Chapter 5: Problems 1-3, 5, 6, 15-19, 23-25, 32-34; Theoretical Exercises 1, 8; Self-Test Problems 1-4 1. [Validity of PDFs] Nine functions f ( u ) are shown below. Note that in each case, f ( u ) = 0 for all u not in the interval specified. In each case, determine whether f ( u ) is a valid probability density function (pdf). If f ( u ) is not a valid pdf, determine if there exists a constant C such that C f ( u ) is a valid pdf. (a) f ( u ) = 2 u, < u < 1. (b) f ( u ) = | u | , | u | < 1 2 (c) f ( u ) = 1- | u | , | u | < 1 , (d) f ( u ) = ln u, < u < 1. Hint: ln u can be integrated by parts. (e) f ( u ) = ln u, < u < 2 , (f) f ( u ) = 2 3 ( u- 1) , < u < 3 , (g) f ( u ) = exp(- 2 u ) , u > 0. (h) f ( u ) = 4exp(- 2 u )- exp(- u ) , u > , (i) f ( u ) = exp(-| u | ) , | u | < 1 , 2. [Calculating probabilities from pdfs] The continuous random variable X has pdf f X ( u ) = ( c (1- u ) , u 1 , , elsewhere . (a) What is the value of c ? (b) Find P { X >- . 5 } . (c) Find P { 6 X 2 > 5 X- 1 } . 3. [Using LOTUS] Let X be a continuous random variable that is uniformly distributed on [- 1 , +1]....
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This note was uploaded on 11/08/2010 for the course ECE ECE 313 taught by Professor S during the Spring '10 term at University of Illinois at Urbana–Champaign.

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PS09 - University of Illinois Fall 2009 ECE 313: Problem...

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