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LECTURE NOTES Course 6.041-6.431 M.I.T. FALL 2000 Introduction to Probability Dimitri P. Bertsekas and John N. Tsitsiklis Professors of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, Massachusetts These notes are copyright-protected but may be freely distributed for instructional nonprofit pruposes. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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