The first editiofl of this book （1986） grew out of a set of notes used by the authorsto teach two one-semester courses on probability and random processes at Rensse-laer Polytechnic Institute （RPI）. At that time the probability course at RPI was re-quired of all students in the Computer and Systems Engineering Program and was a highly recommended elective for students in closely related areas. While many un-dergraduate students took the course in the junior year， many seniors and first-year graduate students took the course for credit as well. Then， as now， most of the stu-dents were engineering students. To serve these students well， we felt that we should be rigorous in introducing fundamental principles while furnishing many op- portunities for students to develop their skills at solving problems.
Introduction to Probability 1.1 INTRODUCTION:WHY STUDY PROBABILITY? 1.2 THE DIFFERENT KINDS OF PROBABILITY A. Probability as Intuition B. Probability as the Ratio of Favorable to Total Outcomes （Classical Theory） C. Probability as a Measure of Frequency of Occurrence D. Probability Based on an Axiomatic Theory 1.3 MISUSES,MISCALCULATIONS,AND PARADOXES 1N PROBABILITY 1.4 SETS,FIELDS,AND EVENTS Examples of Sample Spaces 1.5 AXIOMATIC DEFINITION OF PROBABILITY 1.6 JOINT, CONDITIONAL,AND TOTAL PROBABILITIES;INDEPENDENCE 1.7 BAYES THEOREM AND APPLICATIONS 1.8 COMBINATORICS Occupancy Problems Extensions and Applications 1.9 BERNOULLI TRIALS——BINOMIAL AND MULTINOMIAL PROBABILITY LAWS Multinomial Probability Law 1.10 ASYMPTOTIC BEHAVIOR oF THE BINOMIAL LAW： THE POISSON LAW 1.11 NORMAL APPROXIMATION TO THE BINOMIAL LAW 1.12 SUMMARY PROBLEMS REFERENCES 2 Random Variables 2.1 INTRODUCTION 2.2 DEFINITION OF A RANDOM VARIABLE 2.3 PROBABILITY DISTRIBUTION FUNCTION 2.4 PROBABILITY DENSITY FUNCTION(pdf） Four Other Common Density Functions More Advanced Density Functions 2.5 CONTINUOUS,DISCRETE,AND MIXED RANDOM VARIABLES Examples of Probability Mass Functions 2.6 CONDITIONAL AND JOINT DISTRIBUTIONS AND DENSITIES 2.7 FAILURE RATES 2.8 FUNCTIONS OF A RANDOM VARIABLE 2.9 SOLVING PROBLEMS OF THE TYPE 2.10 SOLVING PROBLEMS OF THE TYPE 2.11 SOLVING PROBLEMS OF THE TYPE W=h(X,Y） 2.12 SUMMARY PROBLEMS REFERENCES ADDITIONAL READING 3. Expectation and Introduction to Estimation 3.1 EXPECTED VALUE OF A RANDOM VARIABLE On the Validity of Equation 3.1 -8 3.2 CONDITIONAL EXPECTATIONS Conditional Expectation as a Random Variable 3.3 MOMENTS Joint Moments Properties of Uncorrelated Random Variables Jointly Gaussian Random Variables Contours of Constant Density of the Joint Gaussian pd{ 3.4 CHEBYSHEV AND SCHWARZ INEQUALITIES Random Variables with Nonnegative Values The Schwarz Inequality 3.5 MOMENT-GENERATING FUNCTIONS 3.6 CHERNOFF BOUND 3.7 CHARACTERISTIC FUNCTIONS Joint Characteristic Functions The Central Limit Theorem 3.8 ESTIMATORS FOR THE MEAN AND VARIANCE OF THE NORMAL LAW Confidence Intervals {or the Mean Confidence Interval for the Variance 3.9 SUMMARY PROBLEMS REFERENCES ADDITIONAL READING 4 Random Vectors and Parameter Estimation 4.1 JOINT DISTRIBUTION AND DENSITIES 4.2 EXPECTATION VECTORS AND COVARIANCE MATRICES 4.3 PROPERTIES OF COVARIANCE MATRICES 4.4 SIMULTANEOUS DIAGONALIZATION OF TWO COVARIANCE MATRICES AND APPLICATIONS IN PATTERN RECOGNITION Projection Maximization of Quadratic Forms 4.5 THE MULTIDIMENSIONAL GAUSSIAN (NORMAL） LAW 4.6 CHARACTERISTIC FUNCTIONS OF RANDOM VECTORS The Characteristic Function of the Gaussian (Normal） Law 4.7 PARAMETER ESTIMATION Estimation of E [ X ] 4.8 ESTIMATION OF VECTOR MEANS AND COVARIANCE MATRICES Estimation of Estimation of the Covariance K 4.9 MAXIMUM LIKELIHOOD ESTIMATORS 4.10 LINEAR ESTIMATION OF VECTOR PARAMETERS 4.11 SUMMARY PROBLEMS REFERENCES ADDITIONAL READING 5 Random Sequences 5.1 BASIC CONCEPTS Infinite-Length Bernoulli Trials Continuity of Probability Measure Statistical Specification of a Random Sequence 5.2 BASIC PRINCIPLES OF DISCRETE-TIME LINEAR SYSTEMS 5.3 RANDOM SEQUENCES AND LINEAR SYSTEMS 5.4 WSS RANDOM SEQUENCES Power Spectral Density Interpretation of the PSD Synthesis of Random Sequences and Discrete-Time Simulation Decimation Interpolation 5.5 MARKOV RANDOM SEQUENCES ARMA Models Markov Chains 5.6 VECTOR RANDOM SEQUENCES AND STATE EQUATIONS 5.7 CONVERGENCE OF RANDOM SEQUENCES 5.8 LAWS OF LARGE NUMBERS 5.9 SUMMARY PROBLEMS REFERENCES 6 Random processes 7 Advanced Topics in Random Processes 8 Applications to Statistical Signal Processing
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