A Certain Uncertainty: Nature's Random Ways

638 Pages·2014·7.056 MB·English

by  Mark P. Silverman

A CERTAIN UNCERTAINTY:
NATURE’S RANDOM WAYS
Based around a series of real-life scenarios, this vivid introduction to statistical
reasoningwillteachyouhowtoapplypowerfulstatistical,qualitative,andprobabil-
istic toolsina technical context.
From analysis of electricity bills, baseball statistics, and the movement of stock
markets, through to the physics of fermions and bosons, and the effects of climate
change, each chapter introduces relevant physical, statistical, and mathematical
principles step-by-step in an engaging narrative style, helping to develop practical
proficiency in theuse of probability and statistical reasoning.
With numerous illustrations, which make it easy to focus on the most important
information,andfull-colorfiguresavailableonlineatwww.cambridge.org/silverman,
thisinsightfulbookisperfectforstudentsandresearchersofanydisciplineinterested
inthe interwoven tapestryof probability, statistics,and physics.
mark p. silverman is the G. A. Jarvis Professor of Physics at Trinity College,
Connecticut. He received his Ph.D. in Chemical Physics from Harvard University,
andhassincepursuedawiderangeofexperimentalandtheoreticalstudiesconcern-
ing the structure of matter, the behavior of light, and the dynamics of stars and
galaxies.

A CERTAIN UNCERTAINTY:
’
NATURE S RANDOM WAYS
MARK P. SILVERMAN
TrinityCollege,Connecticut
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pages cm
Includesbibliographicalreferences.
ISBN978-1-107-03281-1(Hardback)
1. Statisticalphysics. 2. Mathematicalphysics. I. Title.
QC174.8.S5452014
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To Sue,Chrisand Jen
(the onlycertainties inmy life)
Books by Mark P. Silverman
(cid:2) And Yet It Moves: Strange Systems and Subtle Questions in Physics (Cambridge
University Press, 1993)
(cid:2) More Than One Mystery: Explorations in Quantum Interference (Springer, New
York, 1995)
(cid:2) Waves and Grains: Reflections on Light and Learning (Princeton University
Press, 1998)
(cid:2) Probing the Atom: Interactions of Coupled States, Fast Beams, and Loose Electrons
(Princeton UniversityPress, 2000)
(cid:2) A UniverseofAtoms,an Atom inthe Universe (Springer, New York, 2002)
(cid:2) Quantum Superposition: Counterintuitive Consequences of Coherence, Entanglement
and Interference (Springer, Heidelberg, 2008)
Contents
Preface page xiii
Acknowledgments xvii
1 Tools of thetrade 1
1.1 Probability: The calculusof uncertainty 1
1.2 Rules ofengagement 3
1.3 Probability densityfunction and moments 5
1.4 The binomialdistribution: “bits” [Bin(1, p)] and “pieces” [Bin(n,p)] 7
1.5 The Poisson distribution: counting theimprobable 9
1.6 The multinomial distribution: histograms 10
1.7 The Gaussiandistribution: measure ofnormality 12
1.8 The exponential distribution: Waiting forGodot 14
1.9 Moment-generating function 16
1.10 Moment-generatingfunction of alinearcombination of variates 17
1.11 Binomialmoment-generatingfunction 20
1.12 Poisson moment-generating function 22
1.13 Multinomial moment-generating function 24
1.14 Gaussianmoment-generatingfunction 26
1.15 Central Limit Theorem: why things seem mostlynormal 28
1.16 Characteristic function 32
1.17 The uniform distribution 34
1.18 The chi-square (χ2) distribution 38
1.19 Student’st distribution 41
1.20 Inferenceand estimation 45
1.21 The principleof maximum entropy 46
1.22 Shannon entropy function 49
1.23 Entropy and prior information 49
1.24 Method of maximum likelihood 54
1.25 Goodnessof fit: maximum likelihood, chi-square, andP-values 61
1.26 Order and extremes 72
vii
viii Contents
1.27 Bayes’ theorem and themeaning of ignorance 74
Appendices 84
1.28 Rulesof conditionalprobability 84
1.29 Probability density ofa sum ofuniform variates U(0,1) 85
1.30 Probability density ofaχ2variate 86
1.31 Probability density ofthe order statistic Y 87
(i)
1.32 Probability density ofStudent’s tdistribution 89
2 The “fundamentalproblem” ofa practicalphysicist 91
2.1 Bayes’ problem:solution 1(the uniformprior) 91
2.2 Bayes’ problem:solution 2(Jaynes’ prior) 96
2.3 Comparison of thetwo solutions 98
2.4 The Silverman–Bayes experiment 100
2.5 Variationson a theme ofBayes 104
3 “Mother of all randomness” 112
PartI The randomdisintegration of matter 112
3.1 Quantumrandomness: is “the force”withus? 112
3.2 The gamma coincidence experiment 117
3.3 Delusion of layeredhistograms 121
3.4 Elementarystatistics of nuclear decay 122
3.5 Detrendinga time series 128
3.6 Time series: correlations and ergodicity 129
3.7 Periodicity and the sampling theorem 133
3.8 Power spectrum and correlation 138
3.9 Spectralresolution and uncertainty 146
3.10 The non-elementary statistics ofnuclear decay 152
3.11 Recurrence, autocorrelation, andperiodicity 154
3.12 Limitsofdetection 160
3.13 Patternsof randomness: runs 163
3.14 Patternsof randomness: intervals 175
3.15 Final test: intervals, runs, andhistogram shapes 177
3.16 Conclusions andsurprises: the search goes on 181
Appendices 188
3.17 Power spectrumcompleteness relation 188
3.18 Distributions of spectralvariables and autocorrelation functions 189
4 “Mother of all randomness” 194
PartII The randomcreation oflight 194
4.1 The enigma oflight 194
4.2 Quantumvs classical statistics 199
4.3 Occupancy and probability functions 206
4.4 Photon fluctuations 212