1st Edition
Mathematical Models of Information and Stochastic Systems
From ancient soothsayers and astrologists to today’s pollsters and economists, probability theory has long been used to predict the future on the basis of past and present knowledge. Mathematical Models of Information and Stochastic Systems shows that the amount of knowledge about a system plays an important role in the mathematical models used to foretell the future of the system. It explains how this known quantity of information is used to derive a system’s probabilistic properties.
After an introduction, the book presents several basic principles that are employed in the remainder of the text to develop useful examples of probability theory. It examines both discrete and continuous distribution functions and random variables, followed by a chapter on the average values, correlations, and covariances of functions of variables as well as the probabilistic mathematical model of quantum mechanics. The author then explores the concepts of randomness and entropy and derives various discrete probabilities and continuous probability density functions from what is known about a particular stochastic system. The final chapters discuss information of discrete and continuous systems, time-dependent stochastic processes, data analysis, and chaotic systems and fractals.
By building a range of probability distributions based on prior knowledge of the problem, this classroom-tested text illustrates how to predict the behavior of diverse systems. A solutions manual is available for qualifying instructors.
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 Γ 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 Hénon Difference Equation
Single-Particle Single-Well Potential
References
INDEX
Biography
Philipp Kornreich
