Discrete-time stochastic processes

Discrete time stochastic processes

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Unformatted text preview: DISCRETE STOCHASTIC PROCESSES Draft of 2nd Edition R. G. Gallager August 30, 2009 i Contents 1 INTRODUCTION AND REVIEW OF PROBABILITY 1.1 1 3 Assigning probabilities for finite sample spaces . . . . . . . . . . . . 4 The axioms of probability theory . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Axioms for the class of events for a sample space ≠: . . . . . . . . . 6 1.2.2 Axioms of probability . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Probability review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Conditional probabilities and statistical independence . . . . . . . . 8 1.3.2 Repeated idealized experiments . . . . . . . . . . . . . . . . . . . . . 10 1.3.3 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Multiple random variables and conditional probabilities . . . . . . . 12 1.3.5 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.6 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.7 Random variables as functions of other random variables . . . . . . 19 1.3.8 Conditional expectations ........................ 22 1.3.9 Indicator random variables . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.10 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The sample space of a probability model . . . . . . . . . . . . . . . . 1.1.2 1.3 1 1.1.1 1.2 Probability models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 The laws of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4.1 Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4.2 Weak law of large numbers with a finite variance . . . . . . . . . . . 28 1.4.3 Relative frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.4 The central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . 31 ii CONTENTS iii 1.4.5 33 1.4.6 Strong law of large numbers (SLLN) . . . . . . . . . . . . . . . . . . 34 1.4.7 Convergence of random variables . . . . . . . . . . . . . . . . . . . . 39 Relation of probability models to the real world . . . . . . . . . . . . . . . . 42 1.5.1 Relative frequencies in a probability model . . . . . . . . . . . . . . 43 1.5.2 1.5 Weak law with an infinite variance . . . . . . . . . . . . . . . . . . . Relative frequencies in the real world . . . . . . . . . . . . . . . . . . 43 1.5.3 ......... 46 1.5.4 Limitations of relative frequencies . . . . . . . . . . . . . . . . . . . 46 1.5.5 Sub jective probability . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.7.1 Table of standard random variables . . . . . . . . . . . . . . . . . . . 49 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.8 Statistical independence of real-world experiments 2 POISSON PROCESSES 2.1 58 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.1.1 Arrival processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Definition and properties of the Poisson process . . . . . . . . . . . . . . . . 60 2.2.1 Memoryless property . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2.2 Probability density of Sn and S1 , . . . Sn . . . . . . . . . . . . . . . . 64 2.2.3 The PMF for N (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.2.4 Alternate definitions of Poisson processes . . . . . . . . . . . . . . . 67 2.2.5 The Poisson process as a limit of shrinking Bernoulli processes . . . 69 Combining and splitting Poisson processes . . . . . . . . . . . . . . . . . . . 70 2.3.1 Subdividing a Poisson process . . . . . . . . . . . . . . . . . . . . . . 72 2.3.2 Examples using independent Poisson processes . . . . . . . . . . . . 73 2.4 Non-homogeneous Poisson processes . . . . . . . . . . . . . . . . . . . . . . 74 2.5 Conditional arrival densities and order statistics . . . . . . . . . . . . . . . . 77 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.2 2.3 3 RENEWAL PROCESSES 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 92 iv CONTENTS 3.2 Strong Law of Large Numbers for renewal processes . . . . . . . . . . . . . 93 3.3 Expected number of renewals . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.3.1 Laplace transform approach . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.2 Random stopping times . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.3 Wald’s equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Renewal-reward processes; time-averages . . . . . . . . . . . . . . . . . . . . 105 3.4.1 General renewal-reward process...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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