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E-grāmata: Probability for Physicists

  • Formāts: PDF+DRM
  • Sērija : Graduate Texts in Physics
  • Izdošanas datums: 20-May-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319316116
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  • Formāts: PDF+DRM
  • Sērija : Graduate Texts in Physics
  • Izdošanas datums: 20-May-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319316116

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This book is designed as a practical and intuitive introduction to probability, statistics and random quantities for physicists. The book aims at getting to the main points by a clear, hands-on exposition supported by well-illustrated and worked-out examples. A strong focus on applications in physics and other natural sciences is maintained throughout. In addition to basic concepts of random variables, distributions, expected values and statistics, the book discusses the notions of entropy, Markov processes, and fundamentals of random number generation and Monte-Carlo methods.

Recenzijas

irca provides a thorough background; sections cover fundamentals of probability analysis, statistical distributions, and applications of the theories using computer-based algorithms. The work includes useful graphs, four appendixes covering random number generation, and tables of normal distributions. There are extensive references and a valuable index. Summing Up: Recommended. Graduate students; researchers/faculty; professionals. (N. Sadanand, Choice, Vol. 54 (4), December, 2016)

Part I Fundamentals of Probability and Probability Distributions
1 Basic Terminology
3(28)
1.1 Random Experiments and Events
3(3)
1.2 Basic Combinatorics
6(2)
1.2.1 Variations and Permutations
6(1)
1.2.2 Combinations Without Repetition
7(1)
1.2.3 Combinations with Repetition
8(1)
1.3 Properties of Probability
8(3)
1.4 Conditional Probability
11(7)
1.4.1 Independent Events
14(2)
1.4.2 Bayes Formula
16(2)
1.5 Problems
18(13)
1.5.1 Boltzmann, Bose-Einstein and Fermi-Dirac Distributions
18(1)
1.5.2 Blood Types
19(2)
1.5.3 Independence of Events in Particle Detection
21(1)
1.5.4 Searching for the Lost Plane
22(1)
1.5.5 The Monty Hall Problem *
22(3)
1.5.6 Bayes Formula in Medical Diagnostics
25(2)
1.5.7 One-Dimensional Random Walk *
27(2)
References
29(2)
2 Probability Distributions
31(34)
2.1 Dirac Delta
31(4)
2.1.1 Composition of the Dirac Delta with a Function
33(2)
2.2 Heaviside Function
35(1)
2.3 Discrete and Continuous Distributions
36(1)
2.4 Random Variables
37(1)
2.5 One-Dimensional Discrete Distributions
37(2)
2.6 One-Dimensional Continuous Distributions
39(2)
2.7 Transformation of Random Variables
41(4)
2.7.1 What If the Inverse of y = h(x) Is Not Unique?
44(1)
2.8 Two-Dimensional Discrete Distributions
45(2)
2.9 Two-Dimensional Continuous Distributions
47(3)
2.10 Transformation of Variables in Two and More Dimensions
50(6)
2.11 Problems
56(9)
2.11.1 Black-Body Radiation
56(1)
2.11.2 Energy Losses of Particles in a Planar Detector
57(1)
2.11.3 Computing Marginal Probability Densities from a Joint Density
58(2)
2.11.4 Independence of Random Variables in Two Dimensions
60(1)
2.11.5 Transformation of Variables in Two Dimensions
61(2)
2.11.6 Distribution of Maximal and Minimal Values
63(1)
References
64(1)
3 Special Continuous Probability Distributions
65(28)
3.1 Uniform Distribution
65(2)
3.2 Exponential Distribution
67(3)
3.2.1 Is the Decay of Unstable States Truly Exponential?
70(1)
3.3 Normal (Gauss) Distribution
70(4)
3.3.1 Standardized Normal Distribution
71(2)
3.3.2 Measure of Peak Separation
73(1)
3.4 Maxwell Distribution
74(1)
3.5 Pareto Distribution
75(2)
3.5.1 Estimating the Maximum x in the Sample
77(1)
3.6 Cauchy Distribution
77(2)
3.7 The Χ2 distribution
79(1)
3.8 Student's Distribution
79(1)
3.9 F distribution
80(1)
3.10 Problems
80(13)
3.10.1 In-Flight Decay of Neutral Pions
80(3)
3.10.2 Product of Uniformly Distributed Variables
83(1)
3.10.3 Joint Distribution of Exponential Variables
84(1)
3.10.4 Integral of Maxwell Distribution over Finite Range
85(1)
3.10.5 Decay of Unstable States and the Hyper-exponential Distribution
86(3)
3.10.6 Nuclear Decay Chains and the Hypo-exponential Distribution
89(2)
References
91(2)
4 Expected Values
93(30)
4.1 Expected (Average, Mean) Value
93(2)
4.2 Median
95(1)
4.3 Quantiles
96(2)
4.4 Expected Values of Functions of Random Variables
98(2)
4.4.1 Probability Densities in Quantum Mechanics
99(1)
4.5 Variance and Effective Deviation
100(1)
4.6 Complex Random Variables
101(1)
4.7 Moments
102(4)
4.7.1 Moments of the Cauchy Distribution
105(1)
4.8 Two- and d-dimensional Generalizations
106(5)
4.8.1 Multivariate Normal Distribution
110(1)
4.8.2 Correlation Does Not Imply Causality
111(1)
4.9 Propagation of Errors
111(4)
4.9.1 Multiple Functions and Transformation of the Co variance Matrix
113(2)
4.10 Problems
115(8)
4.10.1 Expected Device Failure Time
115(1)
4.10.2 Covariance of Continuous Random Variables
116(1)
4.10.3 Conditional Expected Values of Two-Dimensional Distributions
117(1)
4.10.4 Expected Values of Hyper- and Hypo-exponential Variables
117(2)
4.10.5 Gaussian Noise in an Electric Circuit
119(1)
4.10.6 Error Propagation in a Measurement of the Momentum Vector *
120(1)
References
121(2)
5 Special Discrete Probability Distributions
123(20)
5.1 Binomial Distribution
123(5)
5.1.1 Expected Value and Variance
126(2)
5.2 Multinomial Distribution
128(1)
5.3 Negative Binomial (Pascal) Distribution
129(1)
5.3.1 Negative Binomial Distribution of Order k
129(1)
5.4 Normal Approximation of the Binomial Distribution
130(2)
5.5 Poisson Distribution
132(3)
5.6 Problems
135(8)
5.6.1 Detection Efficiency
135(1)
5.6.2 The Newsboy Problem *
136(2)
5.6.3 Time to Critical Error
138(2)
5.6.4 Counting Events with an Inefficient Detector
140(1)
5.6.5 Influence of Primary Ionization on Spatial Resolution *
140(2)
References
142(1)
6 Stable Distributions and Random Walks
143(34)
6.1 Convolution of Continuous Distributions
143(4)
6.1.1 The Effect of Convolution on Distribution Moments
146(1)
6.2 Convolution of Discrete Distributions
147(2)
6.3 Central Limit Theorem
149(4)
6.3.1 Proof of the Central Limit Theorem
150(3)
6.4 Stable Distributions *
153(2)
6.5 Generalized Central Limit Theorem *
155(1)
6.6 Extreme-Value Distributions *
156(6)
6.6.1 Fisher--Tippett--Gnedenko Theorem
158(1)
6.6.2 Return Values and Return Periods
159(2)
6.6.3 Asymptotics of Minimal Values
161(1)
6.7 Discrete-Time Random Walks *
162(3)
6.7.1 Asymptotics
163(2)
6.8 Continuous-Time Random Walks *
165(2)
6.9 Problems
167(10)
6.9.1 Convolutions with the Normal Distribution
167(1)
6.9.2 Spectral Line Width
168(1)
6.9.3 Random Structure of Polymer Molecules
169(2)
6.9.4 Scattering of Thermal Neutrons in Lead
171(1)
6.9.5 Distribution of Extreme Values of Normal Variables *
172(2)
References
174(3)
Part II Determination of Distribution Parameters
7 Statistical Inference from Samples
177(26)
7.1 Statistics and Estimators
178(6)
7.1.1 Sample Mean and Sample Variance
179(5)
7.2 Three Important Sample Distributions
184(4)
7.2.1 Sample Distribution of Sums and Differences
184(1)
7.2.2 Sample Distribution of Variances
185(1)
7.2.3 Sample Distribution of Variance Ratios
186(2)
7.3 Confidence Intervals
188(4)
7.3.1 Confidence Interval for Sample Mean
188(3)
7.3.2 Confidence Interval for Sample Variance
191(1)
7.3.3 Confidence Region for Sample Mean and Variance
191(1)
7.4 Outliers and Robust Measures of Mean and Variance
192(3)
7.4.1 Chasing Outliers
193(1)
7.4.2 Distribution of Sample Median (and Sample Quantiles)
194(1)
7.5 Sample Correlation
195(3)
7.5.1 Linear (Pearson) Correlation
195(1)
7.5.2 Non-parametric (Spearman) Correlation
196(2)
7.6 Problems
198(5)
7.6.1 Estimator of Third Moment
198(1)
7.6.2 Unbiasedness of Poisson Variable Estimators
199(1)
7.6.3 Concentration of Mercury in Fish
199(2)
7.6.4 Dosage of Active Ingredient
201(1)
References
201(2)
8 Maximum-Likelihood Method
203(24)
8.1 Likelihood Function
203(1)
8.2 Principle of Maximum Likelihood
204(2)
8.3 Variance of Estimator
206(3)
8.3.1 Limit of Large Samples
207(2)
8.4 Efficiency of Estimator
209(3)
8.5 Likelihood Intervals
212(2)
8.6 Simultaneous Determination of Multiple Parameters
214(3)
8.6.1 General Method for Arbitrary (Small or Large) Samples
214(1)
8.6.2 Asymptotic Method (Large Samples)
215(2)
8.7 Likelihood Regions
217(2)
8.7.1 Alternative Likelihood Regions
218(1)
8.8 Problems
219(8)
8.8.1 Lifetime of Particles in Finite Detector
219(2)
8.8.2 Device Failure Due to Corrosion
221(1)
8.8.3 Distribution of Extreme Rainfall
222(2)
8.8.4 Tensile Strength of Glass Fibers
224(1)
References
224(3)
9 Method of Least Squares
227(32)
9.1 Linear Regression
228(14)
9.1.1 Fitting a Polynomial, Known Uncertainties
230(2)
9.1.2 Fitting Observations with Unknown Uncertainties
232(3)
9.1.3 Confidence Intervals for Optimal Parameters
235(1)
9.1.4 How "Good" Is the Fit?
236(1)
9.1.5 Regression with Orthogonal Polynomials *
236(1)
9.1.6 Fitting a Straight Line
237(3)
9.1.7 Fitting a Straight Line with Uncertainties in both Coordinates
240(1)
9.1.8 Fitting a Constant
240(2)
9.1.9 Are We Allowed to Simply Discard Some Data?
242(1)
9.2 Linear Regression for Binned Data
242(3)
9.3 Linear Regression with Constraints
245(3)
9.4 General Linear Regression by Singular-Value Decomposition *
248(1)
9.5 Robust Linear Regression
249(1)
9.6 Non-linear Regression
250(3)
9.7 Problems
253(6)
9.7.1 Two Gaussians on Exponential Background
253(1)
9.7.2 Time Dependence of the Pressure Gradient
254(1)
9.7.3 Thermal Expansion of Copper
255(1)
9.7.4 Electron Mobility in Semiconductor
255(1)
9.7.5 Quantum Defects in Iodine Atoms
256(1)
9.7.6 Magnetization in Superconductor
257(1)
References
257(2)
10 Statistical Tests: Verifying Hypotheses
259(24)
10.1 Basic Concepts
259(5)
10.2 Parametric Tests for Normal Variables
264(5)
10.2.1 Test of Sample Mean
264(1)
10.2.2 Test of Sample Variance
265(1)
10.2.3 Comparison of Two Sample Means, σ2x = σ2y
265(1)
10.2.4 Comparison of Two Sample Means, σ2x ≠ σ2y
266(1)
10.2.5 Comparison of Two Sample Variances
267(2)
10.3 Pearson's Χ2 Test
269(2)
10.3.1 Comparing Two Sets of Binned Data
271(1)
10.4 Kolmogorov--Smirnov Test
271(5)
10.4.1 Comparison of Two Samples
274(1)
10.4.2 Other Tests Based on Empirical Distribution Functions
275(1)
10.5 Problems
276(7)
10.5.1 Test of Mean Decay Time
276(2)
10.5.2 Pearson's Test for Two Histogrammed Samples
278(1)
10.5.3 Flu Medicine
279(1)
10.5.4 Exam Grades
279(1)
References
280(3)
Part III Special Applications of Probability
11 Entropy and Information *
283(24)
11.1 Measures of Information and Entropy
283(5)
11.1.1 Entropy of Infinite Discrete Probability Distribution
285(1)
11.1.2 Entropy of a Continuous Probability Distribution
286(1)
11.1.3 Kullback--Leibler Distance
287(1)
11.2 Principle of Maximum Entropy
288(1)
11.3 Discrete Distributions with Maximum Entropy
289(8)
11.3.1 Lagrange Formalism for Discrete Distributions
289(2)
11.3.2 Distribution with Prescribed Mean and Maximum Entropy
291(1)
11.3.3 Maxwell--Boltzmann Distribution
292(2)
11.3.4 Relation Between Information and Thermodynamic Entropy
294(1)
11.3.5 Bose--Einstein Distribution
295(1)
11.3.6 Fermi--Dirac Distribution
296(1)
11.4 Continuous Distributions with Maximum Entropy
297(1)
11.5 Maximum-Entropy Spectral Analysis
298(9)
11.5.1 Calculating the Lagrange Multipliers
300(2)
11.5.2 Estimating the Spectrum
302(2)
References
304(3)
12 Markov Processes *
307(18)
12.1 Discrete-Time (Classical) Markov Chains
308(5)
12.1.1 Long-Time Characteristics of Markov Chains
309(4)
12.2 Continuous-Time Markov Processes
313(12)
12.2.1 Markov Propagator and Its Moments
314(2)
12.2.2 Time Evolution of the Moments
316(1)
12.2.3 Wiener Process
317(1)
12.2.4 Ornstein--Uhlenbeck Process
318(5)
References
323(2)
13 The Monte--Carlo Method
325(22)
13.1 Historical Introduction and Basic Idea
325(3)
13.2 Numerical Integration
328(5)
13.2.1 Advantage of Monte--Carlo Methods over Quadrature Formulas
332(1)
13.3 Variance Reduction
333(6)
13.3.1 Importance Sampling
333(4)
13.3.2 The Monte--Carlo Method with Quasi-Random Sequences
337(2)
13.4 Markov--Chain Monte Carlo *
339(8)
13.4.1 Metropolis--Hastings Algorithm
339(6)
References
345(2)
14 Stochastic Population Modeling
347(14)
14.1 Modeling Births
347(1)
14.2 Modeling Deaths
348(3)
14.3 Modeling Births and Deaths
351(6)
14.3.1 Equilibrium State
352(1)
14.3.2 General Solution in the Case λn = Nλ, μn = Nμ, λ ≠ μ
353(1)
14.3.3 General Solution in the Case λn = Nλ, μn = Nμ, λ = μ
353(1)
14.3.4 Extinction Probability
353(1)
14.3.5 Moments of the Distribution P(t) in the Case λn = nλ, μn = nμ
354(3)
14.4 Concluding Example: Rabbits and Foxes
357(4)
References
359(2)
Appendix A Probability as Measure * 361(4)
Appendix B Generating and Characteristic Functions + 365(16)
Appendix C Random Number Generators 381(14)
Appendix D Tables of Distribution Quantiles 395(14)
Index 409
Simon Sirca studied physics at the Faculty of Mathematics and Physics, University of Ljubljana, and acquired his first research experience as a young researcher at the Jozef Stefan Institute in Ljubljana and the Institute for Nuclear Physics at the University of Mainz, Germany, concluding his PhD work with the thesis Axial form-factor of the nucleon from coincidence pion electroproduction at low Q2. He was a postdoctoral research associate at the Massachusetts Institute of Technology and the Thomas Jefferson National Accelerator Facility (Jefferson Lab) in the USA. His main research is in the field of hadronic structure and dynamics as explored by scattering of electrons on light nuclei, exploiting state-of-the-art polarized beams, polarized targets, and techniques of recoil polarimetry. He is also involved in theoretical work on quark models of hadrons, with the focus on electroweak processes like pion electroproduction in the nucleon resonance region. He is the head ofthe research group Structure of Hadronic Systems that has been active in the OOPS and BLAST Collaborations at MIT, a collaboration of Jefferson Lab, and the A1 Collaboration and University of Mainz. He is an Associate Professor at the Faculty of Mathematics and Physics, University of Ljubljana, where he has been teaching numerous courses in Mathematical Physics, Modern Physics and Mathematical Physics (Computational Physics).