A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-1 Probability, Statistics, and Random Processes for Electrical Engineering

Chapter 5: Pairs of Random Variables 5.1 Two Random Variables

5.1

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-2 Probability, Statistics, and Random Processes for Electrical Engineering

5.2

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-3 Probability, Statistics, and Random Processes for Electrical Engineering

5.3

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-4 Probability, Statistics, and Random Processes for Electrical Engineering

5.4

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-5 Probability, Statistics, and Random Processes for Electrical Engineering

5.5

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-6 Probability, Statistics, and Random Processes for Electrical Engineering

5.6

5.7

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-7 Probability, Statistics, and Random Processes for Electrical Engineering

5.8

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-8 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-9 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-10 Probability, Statistics, and Random Processes for Electrical Engineering

5.2 Pairs of Discrete Random Variables

5.9

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-11 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-12 Probability, Statistics, and Random Processes for Electrical Engineering

5.10

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-13 Probability, Statistics, and Random Processes for Electrical Engineering

5.11

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-14 Probability, Statistics, and Random Processes for Electrical Engineering

5.12

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-15 Probability, Statistics, and Random Processes for Electrical Engineering

5.13

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-16 Probability, Statistics, and Random Processes for Electrical Engineering

5.14

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-17 Probability, Statistics, and Random Processes for Electrical Engineering

5.15

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-18 Probability, Statistics, and Random Processes for Electrical Engineering

5.3 The Joint pdf of X and Y

5.16

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-19 Probability, Statistics, and Random Processes for Electrical Engineering

5.17

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-20 Probability, Statistics, and Random Processes for Electrical Engineering

5.17

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-21 Probability, Statistics, and Random Processes for Electrical Engineering

5.18

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-22 Probability, Statistics, and Random Processes for Electrical Engineering

5.19

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-23 Probability, Statistics, and Random Processes for Electrical Engineering

5.19

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-24 Probability, Statistics, and Random Processes for Electrical Engineering

5.20

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-25 Probability, Statistics, and Random Processes for Electrical Engineering

5.21

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-26 Probability, Statistics, and Random Processes for Electrical Engineering

5.22

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-27 Probability, Statistics, and Random Processes for Electrical Engineering

5.23

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-28 Probability, Statistics, and Random Processes for Electrical Engineering

5.24

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-29 Probability, Statistics, and Random Processes for Electrical Engineering

5.4 The Joint cdf of Two Continuous Random Variables

5.25

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-30 Probability, Statistics, and Random Processes for Electrical Engineering

5.26

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-31 Probability, Statistics, and Random Processes for Electrical Engineering

5.27

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-32 Probability, Statistics, and Random Processes for Electrical Engineering

5.28

5.28c

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-33 Probability, Statistics, and Random Processes for Electrical Engineering

5.29

5.30

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-34 Probability, Statistics, and Random Processes for Electrical Engineering

5.31

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-35 Probability, Statistics, and Random Processes for Electrical Engineering

5.31

5.32

5.33

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-36 Probability, Statistics, and Random Processes for Electrical Engineering

5.34

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-37 Probability, Statistics, and Random Processes for Electrical Engineering

5.35

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-38 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-39 Probability, Statistics, and Random Processes for Electrical Engineering

5.36

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-40 Probability, Statistics, and Random Processes for Electrical Engineering

5.36b

5.36c

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-41 Probability, Statistics, and Random Processes for Electrical Engineering

5.36c

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-42 Probability, Statistics, and Random Processes for Electrical Engineering

5.5 Independence of Two Random Variables

5.37

5.38

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-43 Probability, Statistics, and Random Processes for Electrical Engineering

5.39

5.40

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-44 Probability, Statistics, and Random Processes for Electrical Engineering

5.41

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-45 Probability, Statistics, and Random Processes for Electrical Engineering

5.42

5.43

5.44

5.45

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-46 Probability, Statistics, and Random Processes for Electrical Engineering

5.46

5.47

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-47 Probability, Statistics, and Random Processes for Electrical Engineering

5.48

5.49

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-48 Probability, Statistics, and Random Processes for Electrical Engineering

5.50

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-49 Probability, Statistics, and Random Processes for Electrical Engineering

5.51

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-50 Probability, Statistics, and Random Processes for Electrical Engineering

5.52

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-51 Probability, Statistics, and Random Processes for Electrical Engineering

5.53

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-52 Probability, Statistics, and Random Processes for Electrical Engineering

5.54

5.55

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-53 Probability, Statistics, and Random Processes for Electrical Engineering

5.6 Joint Moments and Expected Value of a Function of Two Random Variables

5.56

5.57

5.58

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-54 Probability, Statistics, and Random Processes for Electrical Engineering

5.60

5.59

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-55 Probability, Statistics, and Random Processes for Electrical Engineering

5.61

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-56 Probability, Statistics, and Random Processes for Electrical Engineering

5.62

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-57 Probability, Statistics, and Random Processes for Electrical Engineering

5.63

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-58 Probability, Statistics, and Random Processes for Electrical Engineering

5.64

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-59 Probability, Statistics, and Random Processes for Electrical Engineering

5.65

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-60 Probability, Statistics, and Random Processes for Electrical Engineering

5.66

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-61 Probability, Statistics, and Random Processes for Electrical Engineering

5.67

5.68

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-62 Probability, Statistics, and Random Processes for Electrical Engineering

5.69

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-63 Probability, Statistics, and Random Processes for Electrical Engineering

5.70

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-65 Probability, Statistics, and Random Processes for Electrical Engineering

5.71

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-67 Probability, Statistics, and Random Processes for Electrical Engineering

5.72

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-68 Probability, Statistics, and Random Processes for Electrical Engineering

5.73

5.74

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-69 Probability, Statistics, and Random Processes for Electrical Engineering

5.7 Conditional Probability and Conditional Expectation

9

5.75

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-71 Probability, Statistics, and Random Processes for Electrical Engineering

5.76

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-72 Probability, Statistics, and Random Processes for Electrical Engineering

5.77

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-73 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-74 Probability, Statistics, and Random Processes for Electrical Engineering

5.78

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-75 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-76 Probability, Statistics, and Random Processes for Electrical Engineering

5.79

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-77 Probability, Statistics, and Random Processes for Electrical Engineering

5.80

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-78 Probability, Statistics, and Random Processes for Electrical Engineering

5.81

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-79 Probability, Statistics, and Random Processes for Electrical Engineering

5.82

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-80 Probability, Statistics, and Random Processes for Electrical Engineering

5.83

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-81 Probability, Statistics, and Random Processes for Electrical Engineering

5.84

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-82 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-83 Probability, Statistics, and Random Processes for Electrical Engineering

5.85

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-84 Probability, Statistics, and Random Processes for Electrical Engineering

5.86

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-85 Probability, Statistics, and Random Processes for Electrical Engineering

5.87

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-86 Probability, Statistics, and Random Processes for Electrical Engineering

5.88

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-87 Probability, Statistics, and Random Processes for Electrical Engineering

5.89

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-88 Probability, Statistics, and Random Processes for Electrical Engineering

5.8 Functions of Two Random Variables

5.90

5.91

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-89 Probability, Statistics, and Random Processes for Electrical Engineering

5.92

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-90 Probability, Statistics, and Random Processes for Electrical Engineering

5.93

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-91 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-92 Probability, Statistics, and Random Processes for Electrical Engineering

5.95

5.94

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-93 Probability, Statistics, and Random Processes for Electrical Engineering

5.96

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-94 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-95 Probability, Statistics, and Random Processes for Electrical Engineering

(b)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-96 Probability, Statistics, and Random Processes for Electrical Engineering

5.97

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-97 Probability, Statistics, and Random Processes for Electrical Engineering

5.98

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-98 Probability, Statistics, and Random Processes for Electrical Engineering

5.99

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-99 Probability, Statistics, and Random Processes for Electrical Engineering

5.100

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-100 Probability, Statistics, and Random Processes for Electrical Engineering

5.101

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-101 Probability, Statistics, and Random Processes for Electrical Engineering

5.103

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-102 Probability, Statistics, and Random Processes for Electrical Engineering

5.104

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-103 Probability, Statistics, and Random Processes for Electrical Engineering

5.105

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-104 Probability, Statistics, and Random Processes for Electrical Engineering

5.106

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-105 Probability, Statistics, and Random Processes for Electrical Engineering

5.107

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-106 Probability, Statistics, and Random Processes for Electrical Engineering

5.108

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-107 Probability, Statistics, and Random Processes for Electrical Engineering

5.109

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-108 Probability, Statistics, and Random Processes for Electrical Engineering

5.9 Pairs of Jointly Gaussian Random Variables

5.110

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-109 Probability, Statistics, and Random Processes for Electrical Engineering

5.111

5.112

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-110 Probability, Statistics, and Random Processes for Electrical Engineering

5.113

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-111 Probability, Statistics, and Random Processes for Electrical Engineering

5.114

(a)

(b)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-112 Probability, Statistics, and Random Processes for Electrical Engineering

(d)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-113 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-115 Probability, Statistics, and Random Processes for Electrical Engineering

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-116 Probability, Statistics, and Random Processes for Electrical Engineering

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-117 Probability, Statistics, and Random Processes for Electrical Engineering

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-118 Probability, Statistics, and Random Processes for Electrical Engineering

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-119 Probability, Statistics, and Random Processes for Electrical Engineering

5.10 Generating Independent Gaussian Random Variables The following Octave code produces the inverse: function z = rayleigh_rnd(s) u = rand; z = s.*(2.*log(1./(1-u))).^(1/2); end The following Octave generates the requested pairs and plot: len = 10000; X = discrete_rnd([1 -1], [0.5 0.5], 2, len); N = normal_rnd(0, 1, 2, len); Y = X + N; figure; plot(Y(1,:), Y(2,:), "."); Xr = sign(Y); sigerr = (X ~= Xr); biterr = (sigerr(1,:) | sigerr(2,:)); proberr = sum(biterr)./len The following Octave generates the requested pairs and plot: X = normal_rnd(0, 2, 1, 1000); N = normal_rnd(0, 1, 1, 1000); Y = X + N; Xr = Y./(1 + 1/2); err = Xr - X; figure; hist(err, [-3:0.25:3], 4); m = mean(err) v = var(err)

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-120 Probability, Statistics, and Random Processes for Electrical Engineering

The following Octave generates the sequence of Xn and Yn: X = normal_rnd(0, 1, 1, 1000); Y = (X + [0 X(1:999)])./2; figure; plot(Y(1:999), Y(2:1000), "."); Z = (X - [0 X(1:999)])./2; figure; plot(Z(1:999), Z(2:1000), "."); The following Octave generates the specified jointly Gaussian random variables: function vw = gaussian_correlate(xy, mu1, mu2, var1, var2, rho) sig1 = sqrt(var1); sig2 = sqrt(var2); K = [var1, rho*sig1*sig2; rho*sig1*sig2, var2]; [evec eval] = eig(K); A = evec * sqrt(eval); vw = zeros(size(xy)); for i = 1:length(xy) vw(:,i) = A*xy(:,i) + [mu1; mu2]; end end

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-121 Probability, Statistics, and Random Processes for Electrical Engineering

The following Octave generates the specified jointly Gaussian random variables and plot: xy = normal_rnd(0, 1, 2, 1000); vw = gaussian_correlate(xy, 1, -1, 1, 2, -1/2); plot(vw(1,:), vw(2,:), "."); The following Octave generates the specified plot: xy = normal_rnd(0, 1, 2, 1000); muh = 174; muv = 4.4; varh = 42.36; varv = 0.021; covhv = 0.458; sigh = sqrt(varh); sigv = sqrt(varv); rhohv = covhv/(sigh*sigv); xy = normal_rnd(0, 1, 2, 1000); hv = zeros(size(xy)); hv = gaussian_correlate(xy, muh, muv, varh, varv, rhohv); hw = zeros(size(hv)); hw(1,:) = hv(1,:); hw(2,:) = e.^hv(2,:); bmi = hw(2,:)./(hw(1,:).^2); hist(bmi);

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-122 Probability, Statistics, and Random Processes for Electrical Engineering

Problems Requiring Cumulative Knowledge

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-123 Probability, Statistics, and Random Processes for Electrical Engineering

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-126 Probability, Statistics, and Random Processes for Electrical Engineering

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:

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