Statistical Methods in Radiation Physics

This statistics textbook with particular emphasis on radiation protection and dosimetry deals with statistical solutions to problems inherent in health physics measurements and decision making. Dr. Turner begins with a description of our current understanding of the statistical nature of physical processes at the atomic level, including radioactive decay and interactions of radiation with matter. Examples are taken from health physics and the material is presented such that health physicists and most other nuclear professionals will more readily understand the application of the statistical principles due to the familiar context of the examples. Problems are presented at the end of each chapter, with solutions to selected problems provided. In addition, numerous worked examples are included throughout the text.

James S. Bogard is a Senior Health Physicist with Dade Moeller in Oak Ridge, Tennessee, and an Adjunct Professor in the Departments of Nuclear Engineering and Anthropology at the University of Tennessee. He is the author or co-author of over 85 articles, technical reports and presentations, including a workbook of health physics problems and solutions. Dr. Bogard is a past president of the American Academy of Health Physics, a Fellow of the Health Physics Society and a Distinguished Alumnus of Texas State University.

Darryl J. Downing is Vice President, Statistical and Quantitative Sciences, at GlaxoSmithKline Pharmaceutical company. He was previously a member of the research staff at Oak Ridge National Laboratory and led the Statistics Group for 10 of his 20 years at ORNL. Dr. Downing graduated from the University of Florida in 1974 with a Ph.D. in Statistics. He has authored over 50 publications and has been a Fellow of the American Statistical Association since 2002. He is also a member of the International Statistics Institute since 1997 and serves on the editorial board for Pharmaceutical Statistics.

James E. Turner (1930-2008) was a retired Corporate Fellow from Oak Ridge National Laboratory and an Adjunct Professor of Nuclear Engineering at the University of Tennessee. In addition to extensive research and teaching both in the U. S. and abroad, Dr. Turner served on the editorial staffs of several professional journals, including Health Physics and Radiation Research, and was active in a number of scientific organizations. He is a former member of the NCRP, a Past President of the American Academy of Health Physics, and a former Board Member of the Health Physics Society. In 1992 he received the Distinguished Scientific Achievement Award of the Health Physics Society and, in 2000, the William McAdams Outstanding Service Award of the American Board of Health Physics. Dr. Turner published widely in radiation physics and dosimetry and also on the chemical toxicity of metal ions. He is the author of three textbooks.

Preface XIII

1 The Statistical Nature of Radiation, Emission, and Interaction 1

1.1 Introduction and Scope 1

1.2 Classical and Modern Physics – Determinism and Probabilities 1

1.3 Semiclassical Atomic Theory 3

1.4 Quantum Mechanics and the Uncertainty Principle 5

1.5 Quantum Mechanics and Radioactive Decay 8

Problems 11

2 Radioactive Decay 15

2.1 Scope of Chapter 15

2.2 Radioactive Disintegration – Exponential Decay 16

2.3 Activity and Number of Atoms 18

2.4 Survival and Decay Probabilities of Atoms 20

2.5 Number of Disintegrations – The Binomial Distribution 22

2.6 Critique 26

Problems 27

3 Sample Space, Events, and Probability 29

3.1 Sample Space 29

3.2 Events 33

3.3 Random Variables 36

3.4 Probability of an Event 36

3.5 Conditional and Independent Events 38

Problems 45

4 Probability Distributions and Transformations 51

4.1 Probability Distributions 51

4.2 Expected Value 59

4.3 Variance 63

4.4 Joint Distributions 65

4.5 Covariance 71

4.6 Chebyshev’s Inequality 76

4.7 Transformations of Random Variables 77

4.8 Bayes’ Theorem 82

Problems 84

5 Discrete Distributions 91

5.1 Introduction 91

5.2 Discrete Uniform Distribution 91

5.3 Bernoulli Distribution 92

5.4 Binomial Distribution 93

5.5 Poisson Distribution 98

5.6 Hypergeometric Distribution 106

5.7 Geometric Distribution 110

5.8 Negative Binomial Distribution 112

Problems 113

6 Continuous Distributions 119

6.1 Introduction 119

6.2 Continuous Uniform Distribution 119

6.3 Normal Distribution 124

6.4 Central Limit Theorem 132

6.5 Normal Approximation to the Binomial Distribution 135

6.6 Gamma Distribution 142

6.7 Exponential Distribution 142

6.8 Chi-Square Distribution 145

6.9 Student’s t-Distribution 149

6.10 F Distribution 151

6.11 Lognormal Distribution 153

6.12 Beta Distribution 154

Problems 156

7 Parameter and Interval Estimation 163

7.1 Introduction 163

7.2 Random and Systematic Errors 163

7.3 Terminology and Notation 164

7.4 Estimator Properties 165

7.5 Interval Estimation of Parameters 168

7.5.1 Interval Estimation for Population Mean 168

7.5.2 Interval Estimation for the Proportion of Population 172

7.5.3 Estimated Error 173

7.5.4 Interval Estimation for Poisson Rate Parameter 175

7.6 Parameter Differences for Two Populations 176

7.6.1 Difference in Means 176

7.6.1.1 Case 1: 2x and 2y Known 177

7.6.1.2 Case 2: 2x and 2y Unknown, but Equal (¼s2) 178

7.6.1.3 Case 3: 2x and 2y Unknown and Unequal 180

7.6.2 Difference in Proportions 181

7.7 Interval Estimation for a Variance 183

7.8 Estimating the Ratio of Two Variances 184

7.9 Maximum Likelihood Estimation 185

7.10 Method of Moments 189

Problems 194

8 Propagation of Error 199

8.1 Introduction 199

8.2 Error Propagation 199

8.3 Error Propagation Formulas 202

8.3.1 Sums and Differences 202

8.3.2 Products and Powers 202

8.3.3 Exponentials 203

8.3.4 Variance of the Mean 203

8.4 A Comparison of Linear and Exact Treatments 207

8.5 Delta Theorem 210

Problems 210

9 Measuring Radioactivity 215

9.1 Introduction 215

9.2 Normal Approximation to the Poisson Distribution 216

9.3 Assessment of Sample Activity by Counting 216

9.4 Assessment of Uncertainty in Activity 217

9.5 Optimum Partitioning of Counting Times 222

9.6 Short-Lived Radionuclides 223

Problems 226

10 Statistical Performance Measures 231

10.1 Statistical Decisions 231

10.2 Screening Samples for Radioactivity 231

10.3 Minimum Significant Measured Activity 233

10.4 Minimum Detectable True Activity 235

10.5 Hypothesis Testing 240

10.6 Criteria for Radiobioassay, HPS N13.30-1996 248

10.7 Thermoluminescence Dosimetry 255

10.8 Neyman–Pearson Lemma 262

10.9 Treating Outliers – Chauvenet’s Criterion 263

Problems 266

11 Instrument Response 271

11.1 Introduction 271

11.2 Energy Resolution 271

11.3 Resolution and Average Energy Expended per Charge Carrier 275

11.4 Scintillation Spectrometers 276

11.5 Gas Proportional Counters 279

11.6 Semiconductors 280

11.7 Chi-Squared Test of Counter Operation 281

11.8 Dead Time Corrections for Count Rate Measurements 284

Problems 290

12 Monte Carlo Methods and Applications in Dosimetry 293

12.1 Introduction 293

12.2 Random Numbers and Random Number Generators 294

12.3 Examples of Numerical Solutions by Monte Carlo Techniques 296

12.3.1 Evaluation of p ¼ 3.14159265. . . 296

12.3.2 Particle in a Box 297

12.4 Calculation of Uniform, Isotropic Chord Length Distribution in a Sphere 300

12.5 Some Special Monte Carlo Features 306

12.5.1 Smoothing Techniques 306

12.5.2 Monitoring Statistical Error 306

12.5.3 Stratified Sampling 308

12.5.4 Importance Sampling 309

12.6 Analytical Calculation of Isotropic Chord Length Distribution in a Sphere 309

12.7 Generation of a Statistical Sample from a Known Frequency Distribution 312

12.8 Decay Time Sampling from Exponential Distribution 315

12.9 Photon Transport 317

12.10 Dose Calculations 323

12.11 Neutron Transport and Dose Computation 327

Problems 330

13 Dose–Response Relationships and Biological Modeling 337

13.1 Deterministic and Stochastic Effects of Radiation 337

13.2 Dose–Response Relationships for Stochastic Effects 338

13.3 Modeling Cell Survival to Radiation 341

13.4 Single-Target, Single-Hit Model 342

13.5 Multi-Target, Single-Hit Model 345

13.6 The Linear–Quadratic Model 347

Problems 348

14 Regression Analysis 353

14.1 Introduction 353

14.2 Estimation of Parameters b0 and b1 354

14.3 Some Properties of the Regression Estimators 358

14.4 Inferences for the Regression Model 361

14.5 Goodness of the Regression Equation 366

14.6 Bias, Pure Error, and Lack of Fit 369

14.7 Regression through the Origin 375

14.8 Inverse Regression 377

14.9 Correlation 379

Problems 382

15 Introduction to Bayesian Analysis 387

15.1 Methods of Statistical Inference 387

15.2 Classical Analysis of a Problem 388

15.3 Bayesian Analysis of the Problem 390

15.4 Choice of a Prior Distribution 393

15.5 Conjugate Priors 396

15.6 Non-Informative Priors 397

15.7 Other Prior Distributions 401

15.8 Hyperparameters 402

15.9 Bayesian Inference 403

15.10 Binomial Probability 407

15.11 Poisson Rate Parameter 409

15.12 Normal Mean Parameter 414

Problems 419

Appendix 423

Table A.1 Cumulative Binomial Distribution 423

Table A.2 Cumulative Poisson Distribution 424

Table A.3 Cumulative Normal Distribution 426

Table A.4 Quantiles w2 v,a for the chi-squared Distribution with v Degrees of Freedom 429

Table A.5 Quantiles tv,a That Cut off Area a to the Right for Student’s t-distribution with v Degrees of Freedom 431

Table A.6 Quantiles f0.95(v1, v2) for the F Distribution 432

Table A.7 Quantiles f0.99(v1, v2) for the F Distribution 435

References 441

Index 445

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Statistical Methods in Radiation Physics

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