## A Certain Uncertainty**:** Contents

### Preface: An Overview—Start Here

### Chapter 1: Tools of the Trade

1.1 Probability: The Calculus of Uncertainty

1.2 Rules of Engagement

1.3 Probability Density Function and Moments

1.4 The Binomial Distribution: “Bits” [*Bin*(1,*p*)] and “Pieces” [*Bin*(n,*p*)]

1.5 The Poisson Distribution: Counting the Improbable

1.6 The Multinomial Distribution: Histograms

1.7 The Gaussian Distribution: Measure of Normality

1.8 The Exponential Distribution: Waiting for Godot

1.9 Moment-Generating Function

1.10 Moment-Generating Function of a Linear Combination of Variates

1.11 Binomial Moment-Generating Function

1.12 Poisson Moment-Generating Function

1.13 Multinomial Moment-Generating Function

1.14 Gaussian Moment-Generating Function

1.15 Central Limit Theorem: Why Things Seem Mostly Normal

1.16 Characteristic Function

1.17 The Uniform Distribution

1.18 The Chi-Square (*x*^{2}) Distribution

1.19 Student’s *t* Distribution

1.20 Inference and Estimation

1.21 The Principle of Maximum Entropy

1.22 Shannon Entropy Function

1.23 Entropy and Prior Information

1.24 Method of Maximum Likelihood

1.25 Goodness of Fit: Maximum Likelihood, Chi-Square, & *P*-Values

1.26 Order and Extremes

1.27 Bayes’ Theorem and the Meaning of Ignorance

#### APPENDICES

1.28 Rules of Conditional Probability

1.29 Probability Density of a Sum of Uniform Variates *U*(0,1)

1.30 Probability Density of a *x*^{2} Variate

1.31 Probability Density of the Order Statistic *Y*_{(i)}

1.32 Probability Density of Student’s *t* Distribution

### Chapter 2: The “Fundamental Problem” of a Practical Physicist

2.1 Bayes’ Problem: Solution 1 (the Uniform Prior)

2.2 Bayes’ Problem: Solution 2 (Jaynes Prior)

2.3 Comparison of the Two Solutions

2.4 The Silverman-Bayes Experiment

2.5 Variations on a Theme of Bayes

### Chapter 3: “Mother of All Randomness”

#### PART I: The Random Disintegration of Matter

3.1 Quantum Randomness: Is *“The Force”* With Us?

3.2 The Gamma Coincidence Experiment

3.3 Delusion of Layered Histograms

3.4 Elementary Statistics of Nuclear Decay

3.5 Detrending a Time Series

3.6 Time Series: Correlations and Ergodicity

3.7 Periodicity and the Sampling Theorem

3.8 Power Spectrum and Correlation

3.9 Spectral Resolution and Uncertainty

3.10 Non-Elementary Statistics of Nuclear Decay

3.11 Recurrence, Autocorrelation, and Periodicity

3.12 Limits of Detection

3.13 Patterns of Randomness: Runs

3.14 Patterns of Randomness: Intervals

3.15 Final Test: Intervals, Runs, and Histogram Shapes

3.16 Conclusions and Surprises: The Search Goes On

#### APPENDICES

3.17 Power Spectrum Completeness Relation

3.18 Distributions of Spectral Variables and Autocorrelation Functions

### Chapter 4: “Mother of All Randomness”

#### PART II: The Random Creation of Light

4.1 The Enigma of Light

4.2 Quantum vs Classical Statistics

4.3 Occupancy and Probability Functions

4.4 Photon Fluctuations

4.5 The Split-Beam Experiment: Photon Correlations

4.6 Bits, Secrecy, and Photons

4.7 Correlation Experiment with Down-Converted Photons

4.8 Theory of Recurrent Runs

4.9 Runs and the Single Photon: Lessons and Implications

#### APPENDICES

4.10 Chemical Potential of Massless Particles

4.11 Evaluation of Bose-Einstein and Fermi-Dirac Integrals

4.12 Variation in Thermal Photon Energy with Photon Number

4.13 Combinatorial Derivation of the Bose-Einstein Probability

4.14 Generating Function for Probability [Pr(*N _{n} = k*] of

*k*Successes in

*n*trials

### Chapter 5: A Certain Uncertainty

5.1 Beyond the “Beginning of Knowledge”

5.2 Simple Rules: Error Propagation Theory

5.3 Distributions of Products and Quotients

5.4 The Uniform Distribution: Products and Ratios

5.5 The Normal Distribution: Products and Ratios

5.6 Generation of Negative Moments

5.7 Gaussian Negative Moments

5.8 Quantum Test of Composite Measurement Theory

5.9 Cautionary Remarks

5.10 Diagnostic Medical Indices: What do They Signify?

5.11 Secular Equilibrium

5.12 Half-Life Determination by Statistical Sampling: A Mysterious Cauchy Distribution

### APPENDIX

5.13 The Distribution of *W = XY/Z*

### Chapter 6: “Doing the Numbers”: Nuclear Physics and the Stock Market

6.1 The Stock Market is a Casino

6.2 The Details: CREF, AAPL, and GRNG

6.3 Theory of Information *H*

6.4 Is There Information in a Stock Market Time Series?

6.5 Stock Price and Molecular Diffusion

6.6 Random Walk as an Autoregressive Process

6.7 Stocks go UP and UP…and DOWN and DOWN

6.8 What Happened to the Law of Averages?

6.9 Predicting the Future

6.10 Timing is Everything

#### APPENDICES

6.11 Information Inequality *H*(A|B) ≤ *H*(*A*)

6.12 Power Spectral Density of an Autoregressive Time Series

6.13 Exact Maximum Likelihood Estimate of Parameters

### Chapter 7: On Target: Uncertainties of Projectile Flight

7.1 Knowing Where They Come Down

7.2 Distribution of Projectile Ranges

7.3 Energy vs Speed: A Test of Hypotheses

7.4 Play Ball!—Home Runs and Steroids

7.5 Air Resistance

7.6 Theory of Flight

7.7 “Fly(ing) Ball”: Spin and Lift

7.8 Falling Out of the Sky is a Drag

7.9 Descent without Power: How to Rescue a Jumbo Jet Disabled in Flight

#### APPENDICES

7.10 Distribution and Variation of Projectile Range *R*(*V*,Θ)

7.11 Unbiased Estimator of Skewness

### Chapter 8: The Guesses of Groups

8.1 A Radical Hypothesis

8.2 A Mathematical Truism?

8.3 Condorcet’s Jury Theorem

8.4 Epimenides “Paradox of Experts”

8.5 The Silverman GOG Experiments

8.6 Interpretation of the GOG Experiments

8.7 Mining Groups for Information: Galton’s Democratic Model

8.8 Mining Groups for Information: Silverman’s Mixed NU Model

8.9 The BBC-Silverman Experiments: The Reach of Television

8.10 The Log-Normal Distribution: A Fundamental Model of Group Judgment?

8.11 Conclusion: So How “Wise” *Are* Crowds?

#### APPENDICES

8.12 Derivation of the Jury Theorem

8.13 Solution to Logic Problem #1: How Old Are the Children?

8.14 Solution to Logic Problem #2: Where Is the Treasure?

8.15 Origins and Features of a Log-Normal Distribution

### Chapter 9: The Random Flow of Energy

#### PART I: Power to the People

9.1 A Different Kind of Law

9.2 Examining the Data: Time and Autocorrelations

9.3 Examining the Data: Frequency and Power Spectra

9.4 Seeking a Solution: The Construction of Models

9.5 Autoregressive (AR) Time Series

9.6 Moving Average (MA) Time Series

9.7 Combinations: ARMA

9.8 Phase One: Exploration of AR Solutions

9.9 Phase Two: Adaptive and Deterministic Oscillations

9.10 Phase Three: Exploration of MA Solutions

9.11 Phase Four: Judgment—Which Model is Best?

9.12 Electric Shock!

9.13 Two Scenarios: Coincidence or Conspiracy?

#### APPENDICES

9.14 Solution of the AR(12)_{1,12} Master Equation

9.15 Maximum Likelihood Estimate of AR(*n*) Parameters

9.16 Akaike Information Criterion and Log-Likelihood

9.17 Line of Regression to 12-Month Moving Average

### Chapter 10: The Random Flow of Energy

#### PART II: Warning from the Weather Under Ground

10.1 What Lies Above?

10.2 What Lies Beneath?

10.3 Autocorrelation of Underground Temperature

10.4 Fourier Transform and Power Spectrum of Underground Temperature

10.5 Energy Diffusion: Approach I—Deterministic

10.6 Energy Diffusion: Approach II—Stochastic

10.7 Interpreting the Waveforms

10.8 Climate Implications

#### APPENDICES

10.9 Absorption of Solar Radiation by a Sphere

10.10 Autocorrelation of a Decaying Oscillator