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.}

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.. Examples of Random Variable.. Generating Random Variables.. Expectation.

3 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.

4 ( , 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.. Composite Hypotheses.. Example 1.. Example 2.. Example 3.. Summary.. MAP.. MLE.. Hypothesis Test.. Solved Problems.. 1319 Properties.. Linear Least Squares Estimator: LLSE.

5 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.

6 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.. 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.

7 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.

8 Time-Reversibility.. Summary.. Solved Problems.. 24916 Optical Communication Link.. Digital Wireless Communication Link.. 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.

9 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.

10 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.