Statistical Methods in Radiation PhysicsISBN: 9783527411078
466 pages
November 2012

The authors begin 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 problems encountered in health physics, and the material is presented such that health physicists and most other nuclear professionals will more readily understand the application of statistical principles in the familiar context of the examples. Problems are presented at the end of each chapter, with solutions to selected problems provided online. In addition, numerous worked examples are included throughout the text.
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 ChiSquare Distribution 145
6.9 Student’s tDistribution 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 ShortLived 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 N13.301996 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 ChiSquared 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 SingleTarget, SingleHit Model 342
13.5 MultiTarget, SingleHit 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 NonInformative 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 chisquared Distribution with v Degrees of Freedom 429
Table A.5 Quantiles tv,a That Cut off Area a to the Right for Student’s tdistribution 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
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 (19302008) 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.
“This is an excellent, clearly written, statistical textbook with an emphasis on applications to radiological science. . . This book appears to be suitable for this level and should also prove to be a valuable reference for both medical and health physicists.” (Health Physics, 1 March 2014)