Probability theory for quantitative scientists / Luca Leuzzi, Enzo Marinari, Giorgio Parisi.

Format:

Book

Author/Creator:

Leuzzi, Luca, 1972- author.

Marinari, Enzo, author.

Parisi, Giorgio, author.

Language:

English

Subjects (All):

Probabilities.

Quantitative research.

Physical Description:

1 online resource (xiii, 412 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2025.

Summary:

Based on the long-running Probability Theory course at the Sapienza University of Rome, this book offers a fresh and in-depth approach to probability and statistics, while remaining intuitive and accessible in style. The fundamentals of probability theory are elegantly presented, supported by numerous examples and illustrations, and modern applications are later introduced giving readers an appreciation of current research topics. The text covers distribution functions, statistical inference and data analysis, and more advanced methods including Markov chains and Poisson processes, widely used in dynamical systems and data science research. The concluding section, 'Entropy, Probability and Statistical Mechanics' unites key concepts from the text with the authors' impressive research experience, to provide a clear illustration of these powerful statistical tools in action. Ideal for students and researchers in the quantitative sciences this book provides an authoritative account of probability theory, written by leading researchers in the field.

Contents:

Cover

Half-Title Page

Title Page

Imprints Page

Contents

Preface

Guide for Instructors

1 Introduction to Probability

1.1 Definition of Probability

1.2 Basic Properties

1.3 Probability of Set Intersections and Unions

1.4 Conditional Probabilities

1.5 Bayes' Formula

2 Probability Distributions

2.1 Properties of Probability Distributions

2.2 Bernoulli Events and Binomial Distribution

2.3 Poisson Distribution

2.4 Gaussian Distribution

2.5 Cauchy-Lorentz Distribution

Mathematical Appendices

2.A Gaussian Integrals

2.B Euler Gamma Function

2.C Laplace's Method

3 Law of Large Numbers and Central Limit Theorem

3.1 Laws of Large Numbers

3.2 Central Limit Theorem

3.3 Generalized Central Limit Theorem

3.4 Stable Distributions

3.A Markov Limit

3.B Change of Variables

3.C Fourier Transform, Generating Function

3.D Laplace Transform

3.E Hermite Polynomials

3.F Exact Distributions of Sums of Stochastic Variables

3.G Dimensional Analysis

3.H Dirac Delta Distribution Miscellanea

3.I Convergence of Functions

3.J Generation of Pseudo-Random Numbers

4 Large Deviations

4.1 Large Deviations Theorem

4.2 Proof of the Large Deviations Theorem

4.3 Examples of Large Deviations

4.4 Thermodynamic Formalism for Large Deviations

4.5 The Legendre Transform

4.6 Fundamental Theorems on Large Deviations

4.A The Saddle Point Method

5 Statistical Inference and Experimental Data Analysis

5.1 Experimental Data Analysis in the Simple Case

5.2 Use of Bayes' Rule

5.3 Choice of the A Priori Distribution

5.4 General Case of Unknown Probability Distribution

5.5 Resampling Methods

Mathematical Appendix

5.A A Posteriori Distribution of Bernoulli Events.

6 Multivariate and Correlated Experimental Data

6.1 Multivariate Gaussian Data

6.2 Subsampling

6.3 Multivariate Reweighting Method

6.4 Multivariate Resampling Methods

6.5 Least-Squares Method

7 Random Walkers

7.1 Random Walkers in Homogeneous Space

7.2 Random Walkers in Non-Homogeneous Spaces

7.3 Continuum Limit and the Fokker-Planck Equation

7.4 Random Walks with Traps

7.5 Brownian Motion Stationary Solution

7.6 Langevin Equation

7.A Fourier Transform on a Discrete Lattice

7.B Derivation and Properties of the Fokker-Planck Equation

8 Generating Functions and Chain Reactions

8.1 Generating Functions

8.2 Chain Reactions

9 Recurrent Events

9.1 Definitions and Examples

9.2 Classification of Recurrent Events

9.3 Fundamental Relations of Recurrent Events

9.4 Limit Probability Theorem for Recurrent Events

9.A Two Theorems for Divergent Series Summation

10 Markov Chains

10.1 Examples of Markov Chains

10.2 General Properties of Markov Chains

10.3 Further Examples of Markov Chains

10.4 Classification of Markov Chains

10.5 Irreducible Markov Chains Fundamental Theorems

10.6 Balance Equation for Ergodic Irreducible Markov Chains

10.7 Finite Chains and Spectral Decomposition

10.8 Non-Markov Chains

10.A Limit-Sum Exchange under Series Absolute Convergence

10.B Stochastic Matrix Spectral Decomposition

10.C Perron

Frobenius Theorem

11 Numerical Simulations

11.1 Inverse and Reversible Markov Chains

11.2 Detailed Balance for Reversible Markov Chains

11.3 Monte Carlo Method

12 Correlated Events

12.1 Central Limit Distribution for Finite Markov Chains

12.2 Central Limit for Recurrent Events

12.3 Connected Correlation Functions.

12.4 Central Limit and Large Deviations for Correlated Events

12.5 Strongly Correlated Events and Phase Transitions

12.A Asymptotic Behavior of Series and Complex Singularities

13 Continuous-Time Markov Processes

13.1 Poisson Processes

13.2 Pure Birth Processes and Feller's Theorem

13.3 Birth and Death Processes

13.4 Markov Processes

14 Entropy, Probability, and Statistical Mechanics

14.1 Microscopic Entropy and Information Theory

14.2 Entropy in Dynamical Systems

14.3 Intermezzo: Fundamentals of Statistical Mechanics

14.4 Maximum Entropy Principle

14.5 Large Deviations and Thermodynamics

14.6 Configurational Entropy of Glassy Systems

14.A Derivation of Shannon Information and Entropy

14.B Self-Delimiting Messages

References

Index.

Notes:

Title from publisher's bibliographic system (viewed on 30 Jul 2025).

Description based on publisher supplied metadata and other sources.

ISBN:

1-009-58066-3

1-009-58065-5

OCLC:

1530779390