hw03 - University of Illinois Spring 2010 ECE 313: Problem...

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University of Illinois Spring 2010 ECE 313: Problem Set 3: Solutions Discrete Random Variables: pmf, expectation, LOTUS, and variance 1. [ Cumulative distribution function ] Page 174, #4.19 We have that P X = y { } = p y ( ) = F b ( ) ! F b ! ( ) F 0 ( ) ! F 0 ! ( ) = 1 2 ! 0 = 1 2 F 1 ( ) ! F 1 ! ( ) = 3 5 ! 1 2 = 1 10 F 2 ( ) ! F 2 ! ( ) = 4 5 ! 3 5 = 1 5 F 3 ( ) ! F 3 ! ( ) = 9 10 ! 4 5 = 1 10 F 3.5 ( ) ! F 3.5 ! ( ) = 1 ! 9 10 = 1 10 Therefore, p y ( ) = 1/ 2, y = 0 1/10, y = 1 1/ 5, y = 2 1/10, y = 3 1/10, y = 3.5 0, otherwise ! " # # # $ # # # 2. [ Binomial random variable ] Page 176, #4.43 Assume that each digit is interpreted independently. Let E = {3 or more digits are received incorrectly}, and let x = # of incorrectly received bits, then E = {x 3} and x is a binomial random variable with parameters (5, 0.2). ! P E ( ) = 5 k " # $ % k = 3 5 ( 0.2 ( ) k 0.8 ( ) 5 ) k = 5 3 ! " # $ % 0.2 ( ) 3 0.8 ( ) 2 + 5 4 ! " # $ % 0.2 ( ) 4 0.8 ( ) 1 + 5 5 ! " # $ % 0.2 ( ) 5 = 10 0.2 ( ) 3 0.8 ( ) 2 + 5 0.2 ( ) 4 0.8 ( ) 1 + 0.2 ( ) 5 = 0.0579 3. [ Flying used to be fun ] Let X denote the number of passengers that do not
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hw03 - University of Illinois Spring 2010 ECE 313: Problem...

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