In this post, we will see the book Method of Statistical Testing – Monte Carlo Method by Yu. A. Shreider.

# About the book

The present volume of the Mathematical Reference Library is devoted to the so-called method of statistical trials (the Monte Carlo method). In contrast with the earlier volumes, which were devoted to the classical divisions of mathematics and a sharply delineated subject matter with well-established terminology and traditions of exposition, the mathematical methods examined in the present volume have been developed only in the last thirteen years.

These methods, which are applied in the most varied fields of computational mathematics, are unified by a single common idea. They are based on the principle of simulating a statistical experiment by computational techniques and recording the numerical characteristics obtained from this experiment. Therefore, all these methods are united under the common name of statistical trials or the Monte Carlo method. The solution of numerical problems by this method is closer in spirit to physical experiments than to classical computational methods. Error in the Monte Carlo method cannot be sufficiently well evaluated in advance and, as a rule, is found by determining the mean squares for the simulated quantities. In a number of cases the solution cannot be accurately reproduced. The solution is stable with respect to single errors in operation of the given electronic computer.

It is the purpose of the present volume to show the fundamental distinctive features of the Monte Carlo method, giving a sufficiently thorough discussion of the facilities and typical procedures employed and the principal regions of application.

The book was translated from Russian by Scripta Technica and was published in 1964.

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# Contents

PREFACE Vii

## Chapter I. Fundamentals of the Monte Carlo Method. 1

1. Definition and Simple Examples of the Application of the Monte Carlo Method. 1

2. Accuracy of the Monte Carlo Method and its Main

Characteristics. 7

3. Generation of Random Numbers. 13

4. Solution of Systems of Linear Algebraic Equations. 19

5. Random Walk and the Solution of Boundary Value Problems. 23

6. The Monte Carlo Method and the Simulation of Markov Processes in a Computer 31

## Chapter II. Evaluation of Definite Integrals. 39

1. Simple Applications of the Monte Carlo Method. 39

2. Some Methods for Reducing the Variance. 46

3. Evaluation of Multidimensional Integrals. 61

4. Evaluation of Wiener Integrals 73

5. Application of Quasi-Random Points to the Monte Carlo Scheme 78

## Chapter III. Applications of the Monte Carlo Method in Neutron Physics. 84

1. The Monte Carlo Method in Elementary Particle Problems 84

2. Simple Interactions of Neutrons With Nuclei and Their Simulation. 92

3. Passage of Neutrons Through a Plate. 104

4. Some Methods of Calculation of the Criticality of Nuclear Reactors. 118

## Chapter IV. Application of the Monte Carlo Method to the Investigation of Mass Service or Congestion Processes, Including Queueing. 127

1. General Information on Mass-Service Problems. 127

2. Mathematical Description of an Input Consisting of a Stream of Calls Requiring Service 129

3. Mass-Service Systems 134

4. The Generation of Random Streams of Calls 138

5. Structure of an Algorithm for Solving Mass-Service Problems by the Monte Carlo Method. 148

6. Considerations On the Processing of the Results of Simulation. 153

## Chapter V. Application of the Monte Carlo Method to Information Theory 156

1. Statistical Properties of Signals and Noise. 156

2. Formulation of the Basic Problems of Detection Theory. 170

3. Procedure for the Solution of the Main Problems of Detection Theory 184

4. Other Problems 188

## Chapter VI. Generating Uniformly Distributed Random Quantities by Means of Electronic Computers. 196

1. Comparison of Various Methods of Generating Random Quantities. 196

2. Obtaining Uniform Pseudo-Random Number son Computers. 198

3. Criteria to Test the Quality of Uniform Pseudo-Random Numbers. 208

4. Physical Generators of Uniform Random Numbers. 220

5. Tests of the Operation of Random-Number generators. 238

## Chapter VII. Transformation of Random Numbers. 244

1. Properties of Quasi-Uniform Quantities. 244

2. Simulation of Independent Random Events. 247

3. Typical Features of the Simulation of Events by Means of Random Numbers With Few Digits 251

4. Methods of Obtaining Random Numbers With Assigned Distribution Law 252

5. Simulation of Random Vectors and Random Functions 264

6. The Simulation of Certain Multidimensional Quantities 266

APPENDICES. 273

REFERENCES. 281

INDEX. 299

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