E-Book Content
MATHEMATICS RESEARCH DEVELOPMENTS
COMPUTING ALGORITHMS FOR SOLUTIONS OF PROBLEMS IN APPLIED MATHEMATICS AND THEIR STANDARD PROGRAM REALIZATION PART 2 STOCHASTIC MATHEMATICS
K. J. KACHIASHVILI, D. YU. MELIKDZHANIAN AND
A. I. PRANGISHVILI
New York
Copyright © 2015 by Nova Science Publishers, Inc. ISBN: H%RRN
Contents List of Figures
xiii
List of Tables
xv
1 Numerical Methods of Probability Theory and Mathematical Statistics 1.1 Methods of Combinatorics . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Main Concepts and Theorems of Combinatorics . . . . . . . . 1.1.2 Algorithms for the Generation of Numerical Sequences . . . . 1.2 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . 1.2.1 Simplest Discrete Distributions . . . . . . . . . . . . . . . . 1.2.2 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . 1.2.3 Geometric and Pascal Distributions . . . . . . . . . . . . . . 1.2.4 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . 1.2.5 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Series Distribution . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Connection between Different Discrete Distributions . . . . . 1.3 Major Continuous Probability Distributions . . . . . . . . . . . . . . 1.3.1 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Exponential Distribution . . . . . . . . . . . . . . . . . . . . 1.3.3 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Properties of the Normal Distribution Function . . . . . . . . 1.4 m-Dimensional Normal Distribution . . . . . . . . . . . . . . . . . . 1.5 Irregular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Triangular Distribution . . . . . . . . . . . . . . . . . . . . . 1.5.2 Trapezoidal Distribution . . . . . . . . . . . . . . . . . . . . 1.5.3 Generalized Trapezoidal Distribution . . . . . . . . . . . . . 1.5.4 Antimodal-I Distribution . . . . . . . . . . . . . . . . . . . . 1.5.5 Antimodal-II Distribution . . . . . . . . . . . . . . . . . . . 1.6 Basic Probability Distributions Used in Mathematical Statistics . . . . 1.6.1 Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . 1.6.2 Properties of the Chi-Square Distribution Function . . . . . . 1.6.3 Student’s Distribution . . . . . . . . . . . . . . . . . . . . . 1.6.4 Properties of Student’s Distribution Function . . . . . . . . . 1.6.5 Fisher’s Distribution . . . . . . . . . . . . . . . . . . . . . . 1.6.6 Properties of Fisher’s Distribution Function . . . . . . . . . .
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1 2 2 4 8 8 9 12 13 14 15 16 18 18 18 21 22 26 27 27 28 29 29 30 31 31 32 34 35 38 39
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14 1.15
1.6.7 Connection between Different Distributions . . . . . . . . . . . . . Additional Probability Distributions Used in Mathematical Statistics . . . . 1.7.1 Kolmogorov Distribution . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Omega-Square Distribution . . . . . . . . . . . . . . . . . . . . . 1.7.3 D-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Samples and Statistics . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Variational Series . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Estimates of Main Characteristics of a Random Variable . . . . . 1.9.1 Sample Moments and Empiric Probabilities . . . . . . . . . . . . . 1.9.2 Histogram and the Concepts Connected with It . . . . . .