Mathematical models of information and stochastic systems

PREFACE Introduction Historical Development and Aspects of Probability Theory Discussion of the Material in This Text References Events and Density of Events General Probability Concepts Probabilities of Continuous Sets of Events Discrete Events Having the Same Probability Digression of Factorials and the GAMMA Function Continuous Sets of Events Having the Same Probability, Density of States Problems Joint, Conditional, and Total Probabilities Conditional Probabilities Dependent, Independent, and Exclusive Events Total Probability and Bayes' Theorem of Discrete Events Markov Processes Joint, Conditional, and Total Probabilities and Bayes' Theorem of Continuous Events Problems Random Variables and Functions of Random Variables Concept of a Random Variable and Functions of a Random Variable Discrete Distribution Functions Discrete Distribution Functions for More Than One Value of a Random Variable with the Same Probability Continuous Distribution and Density Functions Continuous Distribution Functions for More Than One Value of a Random Variable with the Same Probability Discrete Distribution Functions of Multiple Random Variables Continuous Distribution Functions of Multiple Random Variables Phase Space, a Special Case of Multiple Random Variables Problems Conditional Distribution Functions and a Special Case: The Sum of Two Random Variables Discrete Conditional Distribution Functions Continuous Conditional Distribution Functions A Special Case: The Sum of Two Statistically Independent Discrete Random Variables A Special Case: The Sum of Two Statistically Independent Continuous Random Variables Problems Average Values, Moments, and Correlations of Random Variables and of Functions of Random Variables The Most Likely Value of a Random Variable The Average Value of a Discrete Random Variable and of a Function of a Discrete Random Variable An Often-Used Special Case The Probabilistic Mathematical Model of Discrete Quantum Mechanics The Average Value of a Continuous Random Variable and of a Function of a Continuous Random Variable The Probabilistic Model of Continuous Quantum Mechanics Moments of Random Variables Conditional Average Value of a Random Variable and of a Function of a Random Variable Central Moments Variance and Standard Deviation Correlations of Two Random Variables and of Functions of Random Variables A Special Case: The Average Value of e-jkx References Problems Randomness and Average Randomness The Concept of Randomness of Discrete Events The Concept of Randomness of Continuous Events The Average Randomness of Discrete Events The Average Randomness of Continuous Random Variables The Average Randomness of Random Variables with Values That Have the Same Probability The Entropy of Real Physical Systems and a Very Large Number The Cepstrum Stochastic Temperature and the Legendre Transform Other Stochastic Potentials and the Noise Figure References Problems Most Random Systems Methods for Determining Probabilities Determining Probabilities Based on What Is Known about a System The Poisson Probability and One of Its Applications Continuous Most Random Systems Properties of Gaussian Stochastic Systems Important Examples of Stochastic Physical Systems The Limit of Zero and Very Large Temperatures References Problems Information Information Information in Genes Information Transmission of Discrete Systems Information Transmission of Continuous or Analog Systems The Maximum Information and Optimum Transmission Rates of Discrete Systems The Maximum Information and Optimum Transmission Rates of Continuous or Analog Systems The Bit Error Rate References Problems Random Processes Random Processes Random Walk and the Famous Case of Scent Molecules Emerging from a Perfume Bottle The Simple Stochastic Oscillator and Clocks Correlation Functions of Random Processes Stationarity of Random Processes The Time Average and Ergodicity of Random Processes Partially Coherent Light Rays as Random Processes Stochastic Aspects of Transitions between States Cantor Sets as Random Processes References Problems Spectral Densities Stochastic Power The Power Spectrum and Cross-Power Spectrum The Effects of Filters on the Autocorrelation Function and the Power Spectral Density The Bandwidth of the Power Spectrum Problems Data Analysis Least Square Differences The Special Case of Linear Regression Other Examples Problems Chaotic Systems Fractals Mandelbrot Sets Difference Equations The Henon Difference Equation Single-Particle Single-Well Potential References INDEX