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Unformatted text preview: University of Illinois Spring 2010 ECE 313: Problem Set 10 Due: Wednesday, April 8 at 4 p.m. Reading: Ross, Chapter 5; Lecture Notes 2628. Noncredit Exercises: DO NOT turn these in. Chapter 5: Problems 11,15 and 31; Theoretical Exercises 18,30. This Problem Set contains seven problems. 1. [Moments of random variables] Theoretical Exercises, Chapter 5 of the textbook, Problem 5.5. Solution : Start with E [ X n ] = Z P { X n > t } dt = Z P { X n > x n } d ( x n ) . Since X is nonnegative, P { X n > x n } = P { X > x } . The result follows by observing that d ( x n ) = nx n 1 dx . 2. [Gaussian Random Variables] Let X be a Gaussian random variable with mean = 1 and variance 2 = 4. (a) Find the mean and variance of 2 X + 5. Let ( x ) denote the CDF of a standard Gaussian random variable, and let Q ( x ) = 1 ( x ). Suppose that Calculator A can evaluate only ( x ) and only for nonnegative values of x . On the other hand, suppose that Calculator B can evaluate only Q ( x ), again only for x 0. Both calculators can perform standard functions, like addition and multiplication. For each of the probabilities in parts ( b ) through ( e ), write down two alternative expressions: one for evaluation using Calculator A, and the other for evaluation using Calculator B. (b) P ( X < 0) (c) P ( 10 < X < 5) (d) P (  X  5) (e) P ( X 2 3 X + 2 < 0) Solution: (a) E (2 X + 5] = 2 E [ X ] + 5 = 2 + 5 = 3; var(2 X + 5) = 4var( X ) = 16....
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This note was uploaded on 08/25/2010 for the course ECE 210 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
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