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# lecture_notes_ece5510_f09_all - ECE 5510 Random Processes...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Dr. Neal Patwari University of Utah Department of Electrical and Computer Engineering c circlecopyrt 2009

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ECE 5510 Fall 2009 2 Contents 1 Course Overview 6 2 Events as Sets 7 2.0.1 Set Terminology vs. Probability Terminology . . . . . . . . . . . . . . . . . . 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Important Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Finite, Countable, and Uncountable Event Sets . . . . . . . . . . . . . . . . . . . . . 8 2.3 Operating on Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Disjoint Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Axioms and Properties of Probability 10 3.1 How to Assign Probabilities to Events . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Other Properties of Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Conditional Probability 13 4.1 Conditional Probability is Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Conditional Probability and Independence . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Bayes’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Partitions and Total Probability 16 6 Combinations 17 7 Discrete Random Variables 18 7.1 Probability Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.2 Cumulative Distribution Function (CDF) . . . . . . . . . . . . . . . . . . . . . . . . 21 7.3 Recap of Critical Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7.4 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7.5 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.6 More Discrete r.v.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 8 Continuous Random Variables 25 8.1 Example CDFs for Continuous r.v.s . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 8.2 Probability Density Function (pdf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.3 Expected Value (Continuous) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8.5 Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 9 Method of Moments 29 9.1 Discrete r.v.s Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 9.2 Method of Moments, continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.3 Continuous r.v.s Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 10 Jacobian Method 33
ECE 5510 Fall 2009 3 11 Expectation for Continuous r.v.s 34 12 Conditional Distributions 35 12.1 Conditional Expectation and Probability . . . . . . . . . . . . . . . . . . . . . . . . . 35 13 Joint distributions: Intro (Multiple Random Variables) 37 13.1 Event Space and Multiple Random Variables . . . . . . . . . . . . . . . . . . . . . . 38 13.2 Joint CDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 13.2.1 Discrete / Continuous combinations . . . . . . . . . . . . . . . . . . . . . . . 39 13.3 Joint pmfs and pdfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 13.4 Marginal pmfs and pdfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 13.5 Independence of pmfs and pdfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 13.6 Review of Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 14 Joint Conditional Probabilities 44 14.1 Joint Probability Conditioned on an Event . . . . . . . . . . . . . . . . . . . . . . . 44 14.2 Joint Probability Conditioned on a Random Variable . . . . . . . . . . . . . . . . . . 45 15 Expectation of Joint r.v.s 46 16 Covariance 47 16.1 ‘Correlation’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 16.2 Expectation Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 17 Transformations of Joint r.v.s 49 17.1 Method of Moments for Joint r.v.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 18 Random Vectors 52 18.1 Expectation of R.V.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 19 Covariance of a R.V. 54 20 Joint Gaussian r.v.s 54 20.1 Linear Combinations of Gaussian R.V.s . . . . . . . . . . . . . . . . . . . . . . . . . 56 21 Linear Combinations of R.V.s 56 22 Decorrelation Transformation of R.V.s 59 22.1 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 22.2 Application of SVD to Decorrelate a R.V . . . . . . . . . . . . . . . . . . . . . . . . . 60 22.3 Mutual Fund Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 22.4 Linear Transforms of R.V.s Continued . . . . . . . . . . . . . . . . . . . . . . . . . . 62 23 Random Processes 63 23.1 Continuous and Discrete-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 23.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 23.3 Random variables from random processes . . . . . . . . . . . . . . . . . . . . . . . . 65 23.4 i.i.d. Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 23.5 Counting Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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ECE 5510 Fall 2009 4 23.6 Derivation of Poisson pmf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 23.6.1 Let time interval go to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 24 Poisson Process 68 24.1 Last Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 24.2 Independent Increments Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 24.3 Exponential Inter-arrivals Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 24.4 Inter-arrivals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 24.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 25 Expectation of Random Processes 72 25.1 Expected Value and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 25.2 Autocovariance and Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 25.3 Wide Sense Stationary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 25.3.1 Properties of a WSS Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 26 Power Spectral Density of a WSS Signal 76 27 Review of Lecture 17 77 28 Random Telegraph Wave 79 29 Gaussian Processes 81 29.1 Discrete Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 29.2 Continuous Brownian Motion
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