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  LECTURE NOTESCourse 6.041-6.431M.I.T.FALL 2000 Introduction to Probability  Dimitri P. Bertsekas and John N. Tsitsiklis Professors of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridge, MassachusettsThese notes are copyright-protected but may be freely distributed forinstructional nonprofit pruposes. PDF ebook file resource Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read online Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Where to download Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read file Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf  DOWNLOAD COMPLETE PDF FILE AT https://bookpdf.services/downloads/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf   PDF ebook file resource Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read online Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Where to download Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read file Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf  DOWNLOAD COMPLETE PDF FILE AT https://bookpdf.services/downloads/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf   Contents  1. Sample Space and Probability . . . . . . . . . . . . . . . . 1.1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2. Probabilistic Models . . . . . . . . . . . . . . . . . . . . . . .1.3. Conditional Probability . . . . . . . . . . . . . . . . . . . . .1.4. Independence . . . . . . . . . . . . . . . . . . . . . . . . . .1.5. Total Probability Theorem and Bayes’ Rule . . . . . . . . . . . .1.6. Counting . . . . . . . . . . . . . . . . . . . . . . . . . . .1.7. Summary and Discussion . . . . . . . . . . . . . . . . . . . . 2. Discrete Random Variables . . . . . . . . . . . . . . . . . 2.1. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . .2.2. Probability Mass Functions . . . . . . . . . . . . . . . . . . .2.3. Functions of Random Variables . . . . . . . . . . . . . . . . . .2.4. Expectation, Mean, and Variance . . . . . . . . . . . . . . . . .2.5. Joint PMFs of Multiple Random Variables . . . . . . . . . . . . .2.6. Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . .2.7. Independence . . . . . . . . . . . . . . . . . . . . . . . . . .2.8. Summary and Discussion . . . . . . . . . . . . . . . . . . . . 3. General Random Variables . . . . . . . . . . . . . . . . . 3.1. Continuous Random Variables and PDFs . . . . . . . . . . . . .3.2. Cumulative Distribution Functions . . . . . . . . . . . . . . . .3.3. Normal Random Variables . . . . . . . . . . . . . . . . . . . .3.4. Conditioning on an Event . . . . . . . . . . . . . . . . . . . .3.5. Multiple Continuous Random Variables . . . . . . . . . . . . . .3.6. Derived Distributions . . . . . . . . . . . . . . . . . . . . . .3.7. Summary and Discussion . . . . . . . . . . . . . . . . . . . . 4. Further Topics on Random Variables and Expectations . . . . . . 4.1. Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2. Sums of Independent Random Variables - Convolutions . . . . . . . iii PDF ebook file resource Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read online Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Where to download Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read file Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf  DOWNLOAD COMPLETE PDF FILE AT https://bookpdf.services/downloads/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf   iv  Contents  4.3. Conditional Expectation as a Random Variable . . . . . . . . . . .4.4. Sum of a Random Number of Independent Random Variables . . . .4.5. Covariance and Correlation . . . . . . . . . . . . . . . . . . .4.6. Least Squares Estimation . . . . . . . . . . . . . . . . . . . .4.7. The Bivariate Normal Distribution . . . . . . . . . . . . . . . . 5. The Bernoulli and Poisson Processes . . . . . . . . . . . . . . 5.1. The Bernoulli Process . . . . . . . . . . . . . . . . . . . . . .5.2. The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . 6. Markov Chains . . . . . . . . . . . . . . . . . . . . . . . 6.1. Discrete-Time Markov Chains . . . . . . . . . . . . . . . . . .6.2. Classification of States . . . . . . . . . . . . . . . . . . . . . .6.3. Steady-State Behavior . . . . . . . . . . . . . . . . . . . . . .6.4. Absorption Probabilities and Expected Time to Absorption . . . . .6.5. More General Markov Chains . . . . . . . . . . . . . . . . . . . 7. Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . 7.1. Some Useful Inequalities . . . . . . . . . . . . . . . . . . . . .7.2. The Weak Law of Large Numbers . . . . . . . . . . . . . . . . .7.3. Convergence in Probability . . . . . . . . . . . . . . . . . . . .7.4. The Central Limit Theorem . . . . . . . . . . . . . . . . . . .7.5. The Strong Law of Large Numbers . . . . . . . . . . . . . . . . PDF ebook file resource Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read online Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Where to download Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read file Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf  DOWNLOAD COMPLETE PDF FILE AT https://bookpdf.services/downloads/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf   Preface  These class notes are the currently used textbook for “Probabilistic SystemsAnalysis,” an introductory probability course at the Massachusetts Institute of Technology. The text of the notes is quite polished and complete, but the prob-lems are less so.The course is attended by a large number of undergraduate and graduatestudents with diverse backgrounds. Acccordingly, we have tried to strike a bal-ance between simplicity in exposition and sophistication in analytical reasoning.Some of the more mathematically rigorous analysis has been just sketched orintuitively explained in the text, so that complex proofs do not stand in the wayof an otherwise simple exposition. At the same time, some of this analysis andthe necessary mathematical results are developed (at the level of advanced calcu-lus) in theoretical problems, which are included at the end of the correspondingchapter. The theoretical problems (marked by *) constitute an important com-ponent of the text, and ensure that the mathematically oriented reader will findhere a smooth development without major gaps.We give solutions to all the problems, aiming to enhance the utility of the notes for self-study. We have additional problems, suitable for homeworkassignment (with solutions), which we make available to instructors.Our intent is to gradually improve and eventually publish the notes as atextbook, and your comments will be appreciatedDimitri P. Bertsekasbertsekas@lids.mit.eduJohn N. Tsitsiklis jnt@mit.edu v PDF ebook file resource Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read online Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Where to download Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf|Read file Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf  DOWNLOAD COMPLETE PDF FILE AT https://bookpdf.services/downloads/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf 
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