Probability theory has diverse applications in a plethora of fields, including physics, engineering, computer science, chemistry, biology and economics. This book will familiarize students with various applications of probability theory, stochastic modeling and random processes, using examples from all these disciplines and more. The reader learns via case studies and begins to recognize the sort of problems that are best tackled probabilistically. The emphasis is on conceptual understanding, the development of intuition and gaining insight, keeping technicalities to a minimum. Nevertheless, a glimpse into the depth of the topics is provided, preparing students for more specialized texts while assuming only an undergraduate-level background in mathematics. The wide range of areas covered - never before discussed together in a unified fashion – includes Markov processes and random walks, Langevin and Fokker–Planck equations, noise, generalized central limit theorem and extreme values statistics, random matrix theory and percolation theory.
Author(s): Ariel Amir
Edition: 1
Publisher: Cambridge University Press
Year: 1921
Language: English
Commentary: Vector PDF
Pages: 242
City: Cambridge, UK
Tags: Statistics; Probability Theory; Biology; Markov Models; Randomized Algorithms; Information Theory; Escape Over a Barrier; Noise; Percolation Theory
Cover
Half-title
Title page
Copyright information
Contents
Acknowledgments
1 Introduction
1.1 Probabilistic Surprises
1.2 Summary
1.3 Exercises
2 Random Walks
2.1 Random Walks in 1D
2.2 Derivation of the Diffusion Equation for Random Walks in Arbitrary Spatial Dimension
2.3 Markov Processes and Markov Chains
2.4 Google PageRank: Random Walks on Networks as an Example of a Useful Markov Chain
2.5 Relation between Markov Chains and the Diffusion Equation
2.6 Summary
2.7 Exercises
3 Langevin and Fokker–Planck Equations and Their Applications
3.1 Application of a Discrete Langevin Equation to a Biological Problem
3.2 The Black–Scholes Equation: Pricing Options
3.3 Another Example: The “Well Function” in Hydrology
3.4 Summary
3.5 Exercises
4 Escape Over a Barrier
4.1 Setting Up the Escape-Over-a-Barrier Problem
4.2 Application to the 1D Escape Problem
4.3 Deriving Langer's Formula for Escape-Over-a-Barrier in Any Spatial Dimension
4.4 Summary
4.5 Exercises
5 Noise
5.1 Telegraph Noise: Power Spectrum Associated with a Two-Level-System
5.2 From Telegraph Noise to 1/f Noise via the Superposition of Many Two-Level-Systems
5.3 Power Spectrum of a Signal Generated by a Langevin Equation
5.4 Parseval’s Theorem: Relating Energy in the Time and Frequency Domain
5.5 Summary
5.6 Exercises
6 Generalized Central Limit Theorem and Extreme Value Statistics
6.1 Probability Distribution of Sums: Introducing the Characteristic Function
6.2 Approximating the Characteristic Function at Small Frequencies for Distributions with Finite Variance
6.3 Central Region of CLT: Where the Gaussian Approximation Is Valid
6.4 Sum of a Large Number of Positive Random Variables: Universal Description in Laplace Space
6.5 Application to Slow Relaxations: Stretched Exponentials
6.6 Example of a Stable Distribution: Cauchy Distribution
6.7 Self-Similarity of Running Sums
6.8 Generalized CLT via an RG-Inspired Approach
6.9 Exploring the Stable Distributions Numerically
6.10 RG-Inspired Approach for Extreme Value Distributions
6.11 Summary
6.12 Exercises
7 Anomalous Diffusion
7.1 Continuous Time Random Walks
7.2 Lévy Flights: When the Variance Diverges
7.3 Propagator for Anomalous Diffusion
7.4 Back to Normal Diffusion
7.5 Ergodicity Breaking: When the Time Average and the Ensemble Average Give Different Results
7.6 Summary
7.7 Exercises
8 Random Matrix Theory
8.1 Level Repulsion between Eigenvalues: The Birth of RMT
8.2 Wigner’s Semicircle Law for the Distribution of Eigenvalues
8.3 Joint Probability Distribution of Eigenvalues
8.4 Ensembles of Non-Hermitian Matrices and the Circular Law
8.5 Summary
8.6 Exercises
9 Percolation Theory
9.1 Percolation and Emergent Phenomena
9.2 Percolation on Trees – and the Power of Recursion
9.3 Percolation Correlation Length and the Size of the Largest Cluster
9.4 Using Percolation Theory to Study Random Resistor Networks
9.5 Summary
9.6 Exercises
Appendix A Review of Basic Probability Concepts and Common Distributions
A.1 Some Important Distributions
A.2 Central Limit Theorem
Appendix B A Brief Linear Algebra Reminder, and Some Gaussian Integrals
B.1 Basic Linear Algebra Facts
B.2 Gaussian Integrals
Appendix C Contour Integration and Fourier Transform Refresher
C.1 Contour Integrals and the Residue Theorem
C.2 Fourier Transforms
Appendix D Review of Newtonian Mechanics, Basic Statistical Mechanics, and Hessians
D.1 Basic Results in Classical Mechanics
D.2 The Boltzmann Distribution and the Partition Function
D.3 Hessians
Appendix E Minimizing Functionals, the Divergence Theorem, and Saddle-Point Approximations
E.1 Functional Derivatives
E.2 Lagrange Multipliers
E.3 The Divergence Theorem (Gauss’s Law)
E.4 Saddle-Point Approximations
Appendix F Notation, Notation…
References
Index
