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Contents

Preface ............................................................................ xiii

List of Notations ............................................................... xv

1. Introduction ............................................................. 1 1.1 The Approach ........................................................... 1 1.2 Overview ................................................................... 4 1.3 Instructions ................................................................ 7

2. Experiments and Events ........................................ 9 2.1 Primary Notions ........................................................ 9 2.2 Algebra of Events ..................................................... 12 2.3 Relation of Implication .............................................. 13 2.4 Main Operations with Events .................................... 15 2.5 Main Properties of the Operations with

Events ....................................................................... 18 2.6 Theorem on the Decomposition of an Event

into a Complete Set of Events .................................. 19 2.7 Interpretation of Environmental Phenomena as

Events of Experiments .............................................. 20 2.8 Questions and Exercises .......................................... 23

3. Space of Elementary Events .................................. 25 3.1 Preliminary Remarks ................................................ 25 3.2 Composition of the Space of Elementary

Events ....................................................................... 31

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3.3 Composition of the Space of Elementary Events for Air-quality Monitoring and Forecasting ............... 34

3.4 Characterization of the Eutrophication of a Bay Water ........................................................................ 37

3.5 Questions and Exercises .......................................... 38

4. Probability of Random Events ............................... 41 4.1 Random Events and Random Experiments ............. 41 4.2 The Concept of Probability of a Random

Event ......................................................................... 42 4.3 Adequacy of Chosen Probabilistic Space to the

Given Stochastic Experiment ................................... 43 4.4 Corollaries of Probability Axioms .............................. 45 4.5 Classic Definition of Probability ................................ 47 4.6 Geometric Definition of Probability ........................... 51 4.7 Statistical Definition of Probability ............................ 56 4.8 Questions and Exercises .......................................... 58

5. Conditional Probability and Stochastic Independence: Multistage Probabilistic Evaluation and Forecasting ................................... 61 5.1 Conditional Probability .............................................. 61 5.2 Formula of Total Probability ..................................... 64 5.3 Bayes Formula ......................................................... 64 5.4 Examples of Application ........................................... 65 5.5 Independence of Events ........................................... 68 5.6 Multistage Probabilistic Assessment of

Failure ....................................................................... 70 5.7 Simplified Probabilistic Model for Air-quality

Forecasting ............................................................... 71 5.8 Probability of a Water-purification System

Being Functional ....................................................... 73 5.9 Questions and Exercises .......................................... 73

Contents ix


6. Bernoulli Distribution and Sequences of Independent Trials .................................................. 75 6.1 Bernoulli (Binomial) Distribution ............................... 75 6.2 Sequence of Independent Trials and Its

Mathematical Model ................................................. 76 6.3 Probabilistic Space for a Sequence of

Independent Experiments ........................................ 80 6.4 Bernoulli Scheme of Independent Trials .................. 80 6.5 Examples of Application ........................................... 82 6.6 Application of the Bernoulli Scheme for Air-quality

Assessment .............................................................. 85 6.7 Questions and Exercises .......................................... 86

7. Random Variables and Distribution Functions ................................................................ 89 7.1 Quantities Depending on Random Events ............... 89 7.2 Mathematical Definition of a Random

Variable ..................................................................... 90 7.3 Events Defined by Random Variables ..................... 91 7.4 Independent Random Variables ............................... 92 7.5 Distribution of a Random Variable: the

Distribution Function ................................................. 93 7.6 General Properties of Distribution Functions ........... 93 7.7 Discrete Random Variables ...................................... 95 7.8 Continuous Random Variables ................................ 98 7.9 General Properties of Distribution Density ............... 98 7.10 Distribution Function and Distribution Density

of Functions of Random Variables ........................... 102 7.11 Evaluating Probability of Soil and Groundwater

Contamination ........................................................... 105 7.12 Questions and Exercises .......................................... 108

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8. Numerical Characteristics of Random Variables: Mathematical Expectation, Variance, and Moments of Higher Order .............. 111 8.1 Introduction ............................................................... 111 8.2 Mathematical Expectation of Random

Variables ................................................................... 112 8.3 Statistical Meaning of Mathematical

Expectation ............................................................... 114 8.4 Main Properties of Mathematical Expectation ......... 116 8.5 Functions of Random Variables ............................... 117 8.6 Noncorrelated Random Variables ............................ 119 8.7 Variance of a Random Variable ............................... 120 8.8 Main Properties of Variance ..................................... 121 8.9 Other Characteristics of Dispersion ......................... 121 8.10 Moments of Random Variables of a Higher

Order ......................................................................... 122 8.11 Statistical Linearization ............................................. 123 8.12 Air-quality Comparison ............................................. 125 8.13 Questions and Exercises .......................................... 126

9. Numerical Characteristics of Random Variables: Quantiles ............................................... 129 9.1 Introduction ............................................................... 129 9.2 Probabilistic Meaning and Properties of

Quantiles ................................................................... 131 9.3 Statistical Meaning of Quantiles ............................... 135 9.4 Median, Quartiles, and Other Commonly Used

Quantiles ................................................................... 136 9.5 Application of Quantiles: Minimization of Mean

Losses Caused by Deviation of Random Variables from the Given Level ................................ 138

9.6 Evaluation of Time of Treatment .............................. 141

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9.7 Planning of the Optimal Amount of Oxygen Supply ....................................................................... 141

9.8 Symmetrical Distribution ........................................... 142 9.9 Trace Metal Distribution ............................................ 143 9.10 Questions and Exercises .......................................... 144

10. Probability Distributions: Discrete Case .............. 145 10.1 Binomial (Bernoulli) Distribution ............................... 145 10.2 Numerical Characteristics of Binomial

Distribution ................................................................ 146 10.3 Multistage Processing System: Optimal Stage

Reserve Level ........................................................... 148 10.4 Hypergeometric Distribution ..................................... 149 10.5 Random Selection of Sample Sets from a

Dichotomous Collection ............................................ 152 10.6 Poisson Distribution .................................................. 154 10.7 Poisson Flow of Events ............................................ 157 10.8 Probabilities for the Number of Exceedances .......... 159 10.9 Probabilities of Major Floods .................................... 160 10.10 Questions and Exercises .......................................... 161

11. Probability Distributions: Continuous Case ......... 163 11.1 Introduction ............................................................... 163 11.2 Uniform Distribution .................................................. 163 11.3 Exponential Distribution ............................................ 165 11.4 Normal (Gaussian) Distribution ................................ 169 11.5 Properties of Normal Random Variables ................. 171 11.6 Application of Normal Distribution ............................ 173 11.7 Lognormal Distribution .............................................. 175 11.8 Application of Lognormal Distribution ....................... 177 11.9 Distribution of Solid Particles in Flowing

Water ........................................................................ 179

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11.10 Mean Lifespan of a Bacterium ................................. 180 11.11 Occurrence of Strong Rainfall .................................. 180 11.12 Brownian Motion ....................................................... 181 11.13 Distribution of Grain Sizes ........................................ 181 11.14 Measurements of Trace Levels of Substances:

Normal-lognormal Distribution .................................. 182 11.15 Probabilistic Characterization of a Petroleum

Reservoir ................................................................... 183 11.16 Questions and Exercises .......................................... 188

12. Limit Theorems of the Probability Theory ............ 191 12.1 Introduction ............................................................... 191 12.2 Forms of Convergence for Random

Sequences ................................................................ 192 12.3 Chebyshevs Inequality ............................................. 192 12.4 Law of Large Numbers ............................................. 194 12.5 Central Limit Theorems ............................................ 197 12.6 Practical Use of Central Limit Theorems ................. 198 12.7 Application of Central Limit Theorems to

Bernoullis Scheme ................................................... 199 12.8 Application of Normal Distribution in Biological

Models ...................................................................... 199 12.9 Application of Chebyshevs Inequality ...................... 200 12.10 Maintenance of the Monitoring Stations ................... 202 12.11 Determination of the Number of Tests

Necessary for Confident Decision Making ............... 204 12.12 Questions and Exercises .......................................... 205

13. Probabilistic Decision Making ............................... 207 13.1 Introduction ............................................................... 207 13.2 Risk-assessment Methods ....................................... 208 13.3 Decision Making with Unknown Distributions .......... 210

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13.4 Decision Rules .......................................................... 211 13.5 Reconstruction of a Distribution Function

Based on a Subjective Assessment of Quantiles: Evaluation of the Available Amount of Groundwater Resources of an Aquifer ................. 214

13.6 Investigating Properties of Distribution Functions .................................................................. 218

13.7 Estimation of the Parameters of Distribution ............ 219 13.8 Properties of Good Estimators ................................. 219 13.9 Confidence Interval Construction ............................. 221 13.10 Testing of Hypotheses .............................................. 223 13.11 Air Pollution Investigation ......................................... 225 13.12 Questions and Exercises .......................................... 232

Appendices Appendix 1: Principles of Set Theory ................................. 235 Appendix 2: Methods of Counting ...................................... 245 Appendix 3: Statistical Comparison of an Original

Data Set with Its Subset in Oil Spill Studies ............. 249 Appendix 4: Standard Normal Distribution Function ......... 261

References ..................................................................... 263

Author Index .................................................................. 269

Index ............................................................................... 271

Front MatterPrefaceTable of Contents1. Introduction1.1 The Approach1.2 Overview1.3 Instructions

2. Experiments and Events2.1 Primary Notions2.2 Algebra of Events2.3 Relation of Implication2.4 Main Operations with Events2.5 Main Properties of the Operations with Events2.6 Theorem on the Decomposition of an Event into a Complete Set of Events2.7 Interpretation of Environmental Phenomena as Events of Experiments2.8 Questions and Exercises

3. Space of Elementary Events3.1 Preliminary Remarks3.2 Composition of the Space of Elementary Events3.3 Composition of the Space of Elementary Events for Air-quality Monitoring and Forecasting3.4 Characterization of the Eutrophication of a Bay Water3.5 Questions and Exercises

4. Probability of Random Events4.1 Random Events and Random Experiments4.2 The Concept of Probability of a Random Event4.3 Adequacy of Chosen Probabilistic Space to the Given Stochastic Experiment4.4 Corollaries of Probability Axioms4.5 Classic Definition of Probability4.6 Geometric Definition of Probability4.7 Statistical Definition of Probability4.8 Questions and Exercises

5. Conditional Probability and Stochastic Independence: Multistage Probabilistic Evaluation and Forecasting5.1 Conditional Probability5.2 Formula of Total Probability5.3 Bayes' Formula5.4 Examples of Application5.5 Independence of Events5.6 Multistage Probabilistic Assessment of Failure5.7 Simplified Probabilistic Model for Air-quality Forecasting5.8 Probability of a Water-purification System Being Functional5.9 Questions and Exercises

6. Bernoulli Distribution and Sequences of Independent Trials6.1 Bernoulli (Binomial) Distribution6.2 Sequence of Independent Trials and Its Mathematical Model6.3 Probabilistic Space for a Sequence of Independent Experiments6.4 Bernoulli Scheme of Independent Trials6.5 Examples of Application6.6 Application of the Bernoulli Scheme for Air-quality Assessment6.7 Questions and Exercises

7. Random Variables and Distribution Functions7.1 Quantities Depending on Random Events7.2 Mathematical Definition of a Random Variable7.3 Events Defined by Random Variables7.4 Independent Random Variables7.5 Distribution of a Random Variable: the Distribution Function7.6 General Properties of Distribution Functions7.7 Discrete Random Variables7.8 Continuous Random Variables7.9 General Properties of Distribution Density7.10 Distribution Function and Distribution Density of Functions of Random Variables7.11 Evaluating Probability of Soil and Groundwater Contamination7.12 Questions and Exercises

8. Numerical Characteristics of Random Variables: Mathematical Expectation, Variance, and Moments of Higher Order8.1 Introduction8.2 Mathematical Expectation of Random Variables8.3 Statistical Meaning of Mathematical Expectation8.4 Main Properties of Mathematical Expectation8.5 Functions of Random Variables8.6 Noncorrelated Random Variables8.7 Variance of a Random Variable8.8 Main Properties of Variance8.9 Other Characteristics of Dispersion8.10 Moments of Random Variables of a Higher Order8.11 Statistical Linearization8.12 Air-quality Comparison8.13 Questions and Exercises

9. Numerical Characteristics of Random Variables: Quantiles9.1 Introduction9.2 Probabilistic Meaning and Properties of Quantiles9.3 Statistical Meaning of Quantiles9.4 Median, Quartiles, and Other Commonly Used Quantiles9.5 Application of Quantiles: Minimization of Mean Losses Caused by Deviation of Random Variables from the Given Level9.6 Evaluation of Time of Treatment9.7 Planning of the Optimal Amount of Oxygen Supply9.8 Symmetrical Distribution9.9 Trace Metal Distribution9.10 Questions and Exercises

10. Probability Distributions: Discrete Case10.1 Binomial (Bernoulli) Distribution10.2 Numerical Characteristics of Binomial Distribution10.3 Multistage Processing System: Optimal Stage Reserve Level10.4 Hypergeometric Distribution10.5 Random Selection of Sample Sets from a Dichotomous Collection10.6 Poisson Distribution10.7 Poisson Flow of Events10.8 Probabilities for the Number of Exceedances10.9 Probabilities of Major Floods10.10 Questions and Exercises

11. Probability Distributions: Continuous Case11.1 Introduction11.2 Uniform Distribution11.3 Exponential Distribution11.4 Normal (Gaussian) Distribution11.5 Properties of Normal Random Variables11.6 Application of Normal Distribution11.7 Lognormal Distribution11.8 Application of Lognormal Distribution11.9 Distribution of Solid Particles in Flowing Water11.10 Mean Lifespan of a Bacterium11.11 Occurrence of Strong Rainfall11.12 Brownian Motion11.13 Distribution of Grain Sizes11.14 Measurements of Trace Levels of Substances: Normal-lognormal Distribution11.15 Probabilistic Characterization of a Petroleum Reservoir11.16 Questions and Exercises

12. Limit Theorems of the Probability Theory12.1 Introduction12.2 Forms of Convergence for Random Sequences12.3 Chebyshev's Inequality12.4 Law of Large Numbers12.5 Central Limit Theorems12.6 Practical Use of Central Limit Theorems12.7 Application of Central Limit Theorems to Bernoulli's Scheme12.8 Application of Normal Distribution in Biological Models12.9 Application of Chebyshev's Inequality12.10 Maintenance of the Monitoring Stations12.11 Determination of the Number of Tests Necessary for Confident Decision Making12.12 Questions and Exercises

13. Probabilistic Decision Making13.1 Introduction13.2 Risk-assessment Methods13.3 Decision Making with Unknown Distributions13.4 Decision Rules13.5 Reconstruction of a Distribution Function Based on a Subjective Assessment of Quantiles: Evaluation of the Available Amount of Groundwater Resources of an Aquifer13.6 Investigating Properties of Distribution Functions13.7 Estimation of the Parameters of Distribution13.8 Properties of Good Estimators13.9 Confidence Interval Construction13.10 Testing of Hypotheses13.11 Air Pollution Investigation13.12 Questions and Exercises

AppendicesAppendix 1: Principles of Set TheoryAppendix 2: Methods of CountingAppendix 3: Statistical Comparison of an Original Data Set with Its Subset in Oil Spill StudiesAppendix 4: Standard Normal Distribution Function

ReferencesAuthor IndexIndex

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