Error Analysis with Applications in Engineering (eBook)

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