Error Analysis with Applications in Engineering

Abstract

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 an