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Probability Theory for Quantitative Scientists.pdf
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1 Introduction to Probability 1
1.1 Definition of Probability 1
1.2 Basic Properties 5
1.3 Probability of Set Intersections and Unions 8
1.4 Conditional Probabilities 13
1.5 Bayes’ Formula 15
2 Probability Distributions 23
2.1 Properties of Probability Distributions 23
2.2 Bernoulli Events and Binomial Distribution 28
2.3 Poisson Distribution 32
2.4 Gaussian Distribution 33
2.5 Cauchy–Lorentz Distribution 37
Mathematical Appendices 38
2.A Gaussian Integrals 38
2.B Euler Gamma Function 39
2.C Laplace’s Method 41
3 Law of Large Numbers and Central Limit Theorem 43
3.1 Laws of Large Numbers 44
3.2 Central Limit Theorem 50
3.3 Generalized Central Limit Theorem 56
3.4 Stable Distributions 59
Mathematical Appendices 60
3.A Markov Limit 60
3.B Change of Variables 62
3.C Fourier Transform, Generating Function 63
3.D Laplace Transform 71
3.E Hermite Polynomials 71
3.F Exact Distributions of Sums of Stochastic Variables 72
3.G Dimensional Analysis 75
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vi Contents
3.H Dirac Delta Distribution Miscellanea 78
3.I Convergence of Functions 80
3.J Generation of Pseudo-Random Numbers 84
4 Large Deviations 91
4.1 Large Deviations Theorem 91
4.2 Proof of the Large Deviations Theorem 92
4.3 Examples of Large Deviations 95
4.4 Thermodynamic Formalism for Large Deviations 100
4.5 The Legendre Transform 102
4.6 Fundamental Theorems on Large Deviations 106
Mathematical Appendices 108
4.A The Saddle Point Method 108
5 Statistical Inference and Experimental Data Analysis 117
5.1 Experimental Data Analysis in the Simple Case 117
5.2 Use of Bayes’ Rule 119
5.3 Choice of the A Priori Distribution 124
5.4 General Case of Unknown Probability Distribution 136
5.5 Resampling Methods 146
Mathematical Appendices 159
5.A A Posteriori Distribution of Bernoulli Events 159
6 Multivariate and Correlated Experimental Data 161
6.1 Multivariate Gaussian Data 161
6.2 Subsampling 165
6.3 Multivariate Reweighting Method 165
6.4 Multivariate Resampling Methods 168
6.5 Least-Squares Method 172
7 Random Walkers 188
7.1 Random Walkers in Homogeneous Space 188
7.2 Random Walkers in Non-Homogeneous Spaces 193
7.3 Continuum Limit and the Fokker–Planck Equation 194
7.4 Random Walks with Traps 196
7.5 Brownian Motion Stationary Solution 198
7.6 Langevin Equation 204
Mathematical Appendices 208
7.A Fourier Transform on a Discrete Lattice 208
7.B Derivation and Properties of the Fokker–Planck Equation 211
8 Generating Functions and Chain Reactions 217
8.1 Generating Functions 217
8.2 Chain Reactions 224
vii Contents
9 Recurrent Events 230
9.1 Definitions and Examples 230
9.2 Classification of Recurrent Events 231
9.3 Fundamental Relations of Recurrent Events 233
9.4 Limit Probability Theorem for Recurrent Events 237
Mathematical Appendices 241
9.A Two Theorems for Divergent Series Summation 241
10 Markov Chains 249
10.1 Examples of Markov Chains 249
10.2 General Properties of Markov Chains 251
10.3 Further Examples of Markov Chains 254
10.4 Classification of Markov Chains 259
10.5 Irreducible Markov Chains Fundamental Theorems 266
10.6 Balance Equation for Ergodic Irreducible Markov Chains 273
10.7 Finite Chains and Spectral Decomposition 275
10.8 Non-Markov Chains 281
Mathematical Appendices 283
10.A Limit-Sum Exchange under Series Absolute Convergence 283
10.B Stochastic Matrix Spectral Decomposition 284
10.C Perron--Frobenius Theorem 286
11 Numerical Simulations 289
11.1 Inverse and Reversible Markov Chains 289
11.2 Detailed Balance for Reversible Markov Chains 290
11.3 Monte Carlo Method 294
12 Correlated Events 304
12.1 Central Limit Distribution for Finite Markov Chains 304
12.2 Central Limit for Recurrent Events 308
12.3 Connected Correlation Functions 316
12.4 Central Limit and Large Deviations for Correlated Events 323
12.5 Strongly Correlated Events and Phase Transitions 329
Mathematical Appendices 333
12.A Asymptotic Behavior of Series and Complex Singularities 333
13 Continuous-Time Markov Processes 335
13.1 Poisson Processes 335
13.2 Pure Birth Processes and Feller’s Theorem 337
13.3 Birth and Death Processes 349
13.4 Markov Processes 351
14 Entropy, Probability, and Statistical Mechanics 357
14.1 Microscopic Entropy and Information Theory 358
14.2 Entropy in Dynamical Systems 373
viii Contents
14.3 Intermezzo: Fundamentals of Statistical Mechanics 379
14.4 Maximum Entropy Principle 382
14.5 Large Deviations and Thermodynamics 384
14.6 Configurational Entropy of Glassy Systems 389
Mathematical Appendices 392
14.A Derivation of Shannon Information and Entropy 392
14.B Self-Delimiting Messages 395
References 397
Index 402
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