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
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.
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