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Lecture Notes on Probability Theory and Random Processes

Lecture Notes on Probability Theoryand Random ProcessesJean WalrandDepartment of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeley, CA 94720 August 25, 20042 Table of ContentsTable of Contents3 Abstract9 Introduction11 Modelling Models and Physical Reality.. Concepts and Calculations.. Function of Hidden Variable.. A Look Back.. References.. 122 Probability Choosing At Random .. Events.. Countable Additivity.. Probability Space.. Examples.. Choosing uniformly in{1,2, .. , N}.. Choosing uniformly in [0,1].. Choosing uniformly in [0,1]2.. Summary.. Stars and Bars Method.. Solved Problems.. 193 Conditional Probability and Conditional Probability .. Remark.. Bayes Rule.. Independence.. Example 1.. Example 2.. Definition.. General Definition.. Summary.. Solved Problems.. 324 Random Measurability.. Distribution.

course on probability and random processes in the Department of Electrical Engineering and Computer Sciences at the University of California, Berkeley. The notes do not replace a textbook.

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Transcription of Lecture Notes on Probability Theory and Random Processes

1 Lecture Notes on Probability Theoryand Random ProcessesJean WalrandDepartment of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeley, CA 94720 August 25, 20042 Table of ContentsTable of Contents3 Abstract9 Introduction11 Modelling Models and Physical Reality.. Concepts and Calculations.. Function of Hidden Variable.. A Look Back.. References.. 122 Probability Choosing At Random .. Events.. Countable Additivity.. Probability Space.. Examples.. Choosing uniformly in{1,2, .. , N}.. Choosing uniformly in [0,1].. Choosing uniformly in [0,1]2.. Summary.. Stars and Bars Method.. Solved Problems.. 193 Conditional Probability and Conditional Probability .. Remark.. Bayes Rule.. Independence.. Example 1.. Example 2.. Definition.. General Definition.. Summary.. Solved Problems.. 324 Random Measurability.. Distribution.

2 Examples of Random Variable.. Generating Random Variables.. Expectation.. Function of Random Variable.. Moments of Random Variable.. Inequalities.. Summary.. Solved Problems.. 475 Random Examples.. Joint Statistics.. Independence.. Summary.. Solved Problems.. 756 Conditional Examples.. Example 1.. Example 2.. Example 3.. MMSE.. Two Pictures.. Properties of Conditional Expectation.. Gambling System.. Summary.. Solved Problems.. 957 Gaussian Random Gaussian.. (0,1): Standard Gaussian Random Variable.. ( , 2).. Jointly Gaussian.. (000,III).. Jointly Gaussian.. Conditional Expectation .. Summary.. Solved Problems.. 1088 Detection and Hypothesis Bayesian.. Maximum Likelihood estimation.. Hypothesis Testing Problem.. Simple Hypothesis.. Examples.. Proof of the Neyman-Pearson Theorem.

3 Composite Hypotheses.. Example 1.. Example 2.. Example 3.. Summary.. MAP.. MLE.. Hypothesis Test.. Solved Problems.. 1319 Properties.. Linear Least Squares Estimator: LLSE.. Recursive LLSE.. Sufficient Statistics.. Summary.. LSSE.. Solved Problems.. 14810 Limits of Random Convergence in Distribution.. Transforms.. Almost Sure Convergence.. Example.. Convergence In Probability .. Convergence inL2.. Relationships.. Convergence of Expectation.. 17211 Law of Large Numbers & Central Limit Weak Law of Large Numbers.. Strong Law of Large Numbers.. Central Limit Theorem.. Approximate Central Limit Theorem.. Confidence Intervals.. Summary.. Solved Problems.. 17912 Random Processes Bernoulli - Bernoulli Process.. Time until next 1.. Time since previous 1.. Intervals between 1s.. Saint Petersburg Paradox.. Memoryless Property.. Running Sum.

4 Gamblers Ruin.. Reflected Running Sum.. Scaling: SLLN.. : Brownian.. Poisson Process.. Memoryless Property.. Number of jumps in [0, t].. Scaling: SLLN.. Scaling: Bernoulli Poisson.. Sampling.. Saint Petersburg Paradox.. Stationarity.. Time reversibility.. Ergodicity.. Problems.. 20413 Filtering Linear Time-Invariant Systems.. Definition.. Frequency Domain.. Wide Sense Stationary Processes .. Power Spectrum.. LTI Systems and Spectrum.. Solved Problems.. 22214 Markov Chains - Discrete Definition.. Examples.. Classification.. Invariant Distribution.. First Passage Time.. Time Reversal.. Summary.. Solved Problems.. 23315 Markov Chains - Continuous Definition.. Construction (regular case).. Examples.. Invariant Distribution.. Time-Reversibility.. Summary.. Solved Problems.. 24916 Optical Communication Link.. Digital Wireless Communication Link.

5 M/M/1 Queue.. Speech Recognition.. A Simple Game.. Decisions.. 263A Mathematics Numbers.. Real, Complex, etc.. Min, Max, Inf, Sup.. Summations.. Combinatorics.. Permutations.. Combinations.. Variations.. Calculus.. Sets.. Countability.. Basic Logic.. Proof by Contradiction.. Proof by Induction.. Sample Problems.. 271B Functions275C Nonmeasurable Overview.. Outline.. ConstructingS.. 278D Key Results279E Bertrand s Paradox281F Simpson s Paradox283G Familiar Table.. Examples.. 285 Bibliography293 AbstractThese Notes are derived from lectures and office-hour conversations in a junior/senior-levelcourse on Probability and Random Processes in the Department of Electrical Engineeringand Computer Sciences at the University of California, Notes do not replace a textbook. Rather, they provide a guide through the style is casual, with no attempt at mathematical rigor.

6 The goal to to help the studentfigure out the meaning of various concepts and to illustrate them with choosing a textbook for this course, we always face a dilemma. On the one hand,there are many excellent books on Probability Theory and Random Processes . However, wefind that these texts are too demanding for the level of the course. On the other hand,books written for the engineering students tend to be fuzzy in their attempt to avoid subtlemathematical concepts. As a result, we always end up having to complement the textbookwe select. If we select a math book, we need to help the student understand the meaning ofthe results and to provide many illustrations. If we select a book for engineers, we need toprovide a more complete conceptual picture. These Notes grew out of these efforts at fillingthe will notice that we are not trying to be comprehensive. All the details are availablein textbooks. There is no need to repeat the author wants to thank the many inquisitive students he has had in that class andthe very good teaching assistants, in particular Teresa Tung, Mubaraq Misra, and Eric Chi,who helped him over the years; they contributed many of the reading and keep testing hypotheses!

7 Berkeley, June 2004 -Jean Walrand9 IntroductionEngineering systems are designed to operate well in the face of uncertainty of characteristicsof components and operating conditions. In some case, uncertainty is introduced in theoperations of the system, on how to model uncertainty and how to analyze its effects is or should be an essential part of an engineer s education. Randomness is a key element of all systemswe design. Communication systems are designed to compensate for noise. Internet routersare built to absorb traffic fluctuations. Building must resist the unpredictable vibrationsof an earthquake. The power distribution grid carries an unpredictable load. Integratedcircuit manufacturing steps are subject to unpredictable variations. Searching for genes islooking for patterns among unknown should you understand about Probability ? It is a complex subject that has beenconstructed over decades by pure and applied mathematicians. Thousands of books explorevarious aspects of the Theory .

8 How much do you really need to know and where do youstart?The first key concept is how to model uncertainty (see Chapter2-3). What do we meanby a Random experiment? Once you understand that concept, the notion of a randomvariable should become transparent (see Chapters4-5). You may be surprised to learn thata Random variable does not vary! Terms may be confusing. Once you appreciate the notionof randomness, you should get some understanding for the idea of expectation ( )and how observations modify it (Chapter6). A special class of Random variables (Gaussian)12are particularly useful in many applications (Chapter7). After you master these key notions,you are ready to look at detection (Chapter8) and estimation problems (Chapter9). Theseare representative examples of how one can process observation to reduce uncertainty. Thatis, how one learns. Many systems are subject to the cumulative effect of many sources ofrandomness. We study such effects in Chapter11after having provided some backgroundin Chapter10.

9 The final set of important notions concern Random Processes : uncertainevolution over time. We look at particularly useful models of such Processes in Chapters12-15. We conclude the Notes by discussing a few applications in concepts are difficult, but the math is not (Appendix??reviews what you shouldknow). The trick is to know what we are trying to compute. Look at examples and inventnew ones to reinforce your understanding of ideas. Don t get discouraged if some ideas seemobscure at first, but do not let the obscurity persist! This stuff is not that hard, it is onlynew for 1 Modelling UncertaintyIn this chapter we introduce the concept of a model of an uncertain physical system. Westress the importance of concepts that justify the structure of the Theory . We comment onthe notion of a hidden variable. We conclude the chapter with a very brief historical lookat the key contributors and some Notes on Models and Physical RealityProbability Theory is a mathematical model of uncertainty.

10 In these Notes , we introduceexamples of uncertainty and we explain how the Theory models is important to appreciate the difference between uncertainty in the physical worldand the models of Probability Theory . That difference is similar to that between laws oftheoretical physics and the real world: even though mathematicians view the Theory asstanding on its own, when engineers use it, they see it as a model of the physical flipping a fair coin repeatedly. Designate by 0 and 1 the two possible outcomesof a coin flip (say 0 for head and 1 for tail). This experiment takes place in the physicalworld. The outcomes are uncertain. In this chapter, we try to appreciate the probabilitymodel of this experiment and to relate it to the physical 1. MODELLING Concepts and CalculationsIn our many years of teaching Probability models, we have always found that what ismost subtle is the interpretation of the models, not the calculations. In particular, thisintroductory course uses mostly elementary algebra and some simple calculus.


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