Synopses & Reviews
This book presents, in the simplest possible manner, those branches of error analysis which find direct applications in solving various problems in engineering. Chapters I, II, III, and IV contain a presentation of the fundamentals of error calculus: basic characteristics of error distributions, histograms and their various applications, basic continuous distributions of errors and functions of independent random variables. In Chapter V, two-dimensional distributions of errors are discussed with applications. Fundamentals of the theory of two-dimensional continuous independent and dependent random variables are also discussed in that chapter. Then the methods of determination of the ellipses of probability concentration for the two-dimensional continuous normal distribution are given. Chapter VI deals with two-dimensional vectorial functions of independent random variables along with practical applications to the analysis of the positioning accuracy of mechanisms with two-dimensional movements. The procedure of determination of ellipses of probability concentration is also described. In Chapter VII, three-dimensional distributions of errors are considered, while Chapter VIII deals with the three-dimensional vectorial functions of independent random variables. The theory is illustrated by examples of the analysis of the positioning accuracy of robot manipulators. The examples of determining the ellipsoids of probability concentration are presented. Chapter IX contains error analysis-inspired problems that are described by implicit equations and Chapter X presents useful definitions and facts of probability theory for future readings. This book has been written for readers whose main interests are applications of error calculus in various problems of engineering. In all ten chapters much attention is paid to the practical significance of error analysis.
Table of Contents
1 Basic characteristics of error distribution; histograms 1 .I Introductory remarks; histograms; 1.2 The average of a sample of measurements ; 1.3 Dispersion measures in error analysis ; 1.4 Cumulative frequency distribution ; 1.5 Examples of empirical distributions ; 1.6 Parameters obtained from the measured data and their theoretical values ; Problems ; References ; 2 Random variables and probability; normal distribution ; 2.1 Probability and random variables ; 2.2 The cumulative distribution function; the probability density function; 2.3 Moments; 2.4 The normal probability distribution ; 2.5 Two-dimensional gravity flow of granular material ; Problems; References ; 3 Probability distributions and their characterizations ; 3.1 The characteristic function of a distribution ; 3.2 Constants characterizing the random variables ; 3.3 Deterministic functions of random variabies; 3.4 Some other one-dimensionat distributions ; 3.4.1 Discrete probability distributions ; 3.4.2 Corttinuous probability distribuhoas ; 3.4.3 Remarks on other probability distributions ; 3.4.4 Measures of deviation from the normal distribution ; 3.5 Approximate methods for constructing a probability density function ; 3.6 Multi-dimensional probability distributions ; Problerns ; References ; 4 Functions of independent random variables ; 4.1 Basic relations.. ; 4.2 Simple examples of applications ; 4.3 Examples of applications in non-direct measuremenis ; 4.4 Remarks on applications in the calculus of tolerance limits ; 4.5 Statical analogy in the analysis of complex dimension nets ; Problems ; References ; 5 Two-dimensional Distributions ; 5.1 Introductory remarks ; 5.2 Linear regression of experimental observations ; 5.2.1 Nonparametric regression ; 5.2.2 The method of least squares for determining the linear regression line ; 5.2.3 The method of moments for determining the linear regression line l 5.3 Litlear correlation between experimentally determined quantities ; 5.4 Two-dimensional continuous random variables ; 5.5 The two-dimensional normal distribution ; 5.5.1 The case of independent random variables ; 5.5.2 The circular normal distribution ; 5 5.3 Three-dimensional gravity flow of granular media ; 5.5.4 The case of dependent randoin variables ; Problems ; References ; 6 Two-dimeosional functions of independent random variables ; 6 .1 Basic relations ; 6.2 The rectangillar distribution of independent random variables ; 6.2.1 Analytical method for determining two-dimensional tolerance limits polygons ; .2.2 Statical analogy method for determining two-dimensional tolerance limit polygons ; 6.2.3 Graphical method for determining two-dimensional tolerance limits polygon . Williot's diagram ; 6.3 The normal distribution of independent random variables ; 6.4 Indirect deternlination of the ellipses of probability concentration; Problems ; References ; 7 Three-dimensional distributions; 7.1 General remarks; 7.2 Continuous three-dimensional random variables ; 7.3 Thc three-dimensional normal distribution ; 7.3.1 Independent random variables ; 7.3.2 The spherical normd distribution ; 7.3.3 The case of dependent random variables ; Problems ; References ; 8 Three-dimensional functions of independent random variables ; 8.1 Basic relations ; 8.2 The rectangular distribution of independent random variables ; 8.3 The normal distribution of independent random variables ; 8.4 Indirect determination of the ellipsoids of probability concentmtion; Problems ; References; 9 Problems described by implicit equations: 9.1 Introduction ; 9.2 Statistically independent random variables ; 9.2.1 Two independent random variables; 9.2.2 A function of independent random variables ; 9.3 Statistically dependent random variables ; 9.3.1 Two dependent random variables ; 9.3.2 The case of Gaussian random variables ; 9.3.3 More random variables: the Rosenblatt transformation ; 9.4 Computational problems ; References ; 10 Useful definitions and facts of probability theory for further reading ; 10.1 Statistical linearization ; 10.2 Multi-dimension regression ; 10.3 Limit theorems of probability theory ; 10.3.1 Concepts of probabilistic convergence ; 10.3.2 The law of large numbers ; 10.3.3 The central limit kheurerns ; 10.4 Elements of rnathematicaI siatisiics ; 10.4.1 Estimators ; 10.4.2 Testing statistical hypotheses; 10.4.3 Confidence intervals; 10.5 Bibliographical notes for future studies; References; Index; Solutions.