Error Analysis with Applications in Engineering

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.