lecture_notes_ece5510_f09_all

lecture_notes_ece5510_f09_all - ECE 5510: Random Processes...

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