Bertsekas Dimitri, Tsitsiklis John N. - Introduction To Probability.pdf

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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 . . . . . . . . . . . . . . . . . . . .
4. Further Topics on Random Variables and Expectations . . . . . .
4.1. Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Sums of Independent Random Variables - Convolutions . . . . . . .
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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 . . . . . . . . . . . . . . . .

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