Introductory Statistics and Random Phenomena

by Denker, Manfred; Woyczynski, Wojbor A.; Ycart, Bernard

• ISBN13: 9780817640316
• ISBN10: 0817640312
• Format: Hardcover
• Copyright: 1999-03-01
• Publisher: Birkhauser

Table of Contents

Preface	xi	(4)
Introduction	xv	(8)
Notation and Abbreviations	xxiii
I DESCRIPTIVE STATISTICS-COMPRESSING DATA	1	(200)
1 Why One Needs to Analyze Data	3	(52)
1.1 Coin tossing, lottery, and the stock market	3	(6)
1.2 Inventory problems in management	9	(1)
1.3 Battery life and quality control in manufacturing	10	(2)
1.4 Reliability of complex systems	12	(5)
1.5 Point processes in time and space	17	(4)
1.6 Polls-social sciences	21	(5)
1.7 Time series	26	(3)
1.8 Repeated experiments and testing	29	(3)
1.9 Simple chaotic dynamical systems	32	(7)
1.10 Complex dynamical systems	39	(2)
1.11 Pseudorandom number generators and the Monte-Carlo methods	41	(4)
1.12 Fractals and image reconstruction	45	(1)
1.13 Coding and decoding, unbreakable ciphers	46	(3)
1.14 Experiments, exercises, and projects	49	(3)
1.15 Bibliographical notes	52	(3)
2 Data Representation and Compression	55	(64)
2.1 Data types, categorical data	55	(8)
2.2 Numerical data: order statistics, median, quantiles	63	(7)
2.3 Numerical data: histograms, means, moments	70	(7)
2.4 Location, dispersion, and shape parameters	77	(5)
2.5 Probabilities: a frequentist viewpoint	82	(6)
2.6 Multidimensional data: histograms and other graphical representations	88	(4)
2.7 2-D data: regression and correlations	92	(7)
2.8 Fractal data	99	(6)
2.9 Measuring information content: entropy	105	(6)
2.10 Experiments, exercises, and projects	111	(4)
2.11 Bibliographical notes	115	(4)
3 Analytic Representation of Random Experimental Data	119	(82)
3.1 Repeated experiments and the law of large numbers	119	(9)
3.2 Characteristics of experiments: distribution functions, densities, means, variances	128	(8)
3.3 Uniform distributions, simulation of random quantities, the Monte Carlo method	136	(3)
3.4 Bernoulli and binomial distributions	139	(6)
3.5 Rescaling probabilities: Poisson approximation	145	(7)
3.6 Stability of Fluctuations Law: Gaussian approximation	152	(11)
3.7 How to estimate p in Bernoulli experiments	163	(8)
3.8 Other continuous distributions; Gamma function calculus	171	(14)
3.9 Testing the fit of a distribution	185	(3)
3.10 Random vectors and multivariate distributions	188	(8)
3.11 Experiments, exercises, and projects	196	(2)
3.12 Bibliographical notes	198	(3)
II MODELING UNCERTAINTY	201	(164)
4 Algorithmic Complexity and Random Strings	203	(40)
4.1 Heart of randomness: when is random -- random?	203	(4)
4.2 Computable strings and the Turing machine	207	(5)
4.3 Kolmogorov complexity and random strings	212	(6)
4.4 Typical sequences: Martin-Lof tests of randomness	218	(8)
4.5 Stability of subsequences: von Mises randomness	226	(2)
4.6 Computable framework of randomness: degrees of irregularity	228	(9)
4.7 Experiments, exercises, and projects	237	(3)
4.8 Bibliographical notes	240	(3)
5 Statistical Independence and Kolmogorov's Probability Theory	243	(50)
5.1 Description of experiments, random variables, and Kolmogorov's axioms	243	(15)
5.2 Uniform discrete distributions and counting	258	(3)
5.3 Statistical independence as a model for repeated experiments	261	(4)
5.4 Expectations and other characteristics of random variables	265	(11)
5.4.1 Expectations	265	(2)
5.4.2 Expectations of functions of random variables. Variance	267	(1)
5.4.3 Expectations of functions of vectors. Covariance	268	(1)
5.4.4 Expectation of the product. Variance of the sum of independent random variables	269	(3)
5.4.5 Moments and the moment generating function	272	(3)
5.4.6 Expectations of general random variables	275	(1)
5.5 Averages of independent random variables	276	(5)
5.6 Laws of large numbers and small deviations	281	(3)
5.7 Central limit theorem and large deviations	284	(3)
5.8 Experiments, exercises, and projects	287	(5)
5.9 Bibliographical Notes	292	(1)
6 Chaos in Dynamical Systems: How Uncertainty Arises in Scientific and Engineering Phenomena	293	(72)
6.1 Dynamical systems: general concepts and typical examples	293	(14)
6.2 Orbits and fixed points	307	(18)
6.3 Stability of frequencies and the ergodic theorem	325	(15)
6.4 Stability of fluctuations and the central limit theorem	340	(9)
6.5 Attractors, fractals, and entropy	349	(11)
6.6 Experiments, exercises, and projects	360	(2)
6.7 Bibliographical notes	362	(3)
III MODEL SPECIFICATION-DESIGN OF EXPERIMENTS	365	(138)
7 General Principles of Statistical Analysis	367	(26)
7.1 Design of experiments and planning of investigation	367	(2)
7.2 Model selection	369	(3)
7.3 Determining the method of statistical inference	372	(12)
7.3.1 Maximum likelihood estimator (MLE)	373	(3)
7.3.2 Least squares estimator (LSE)	376	(5)
7.3.3 Method of moments (MM)	381	(2)
7.3.4 Concluding remarks	383	(1)
7.4 Estimation of fractal dimension	384	(3)
7.5 Practical side of data collection and analysis	387	(2)
7.6 Experiments, exercises, and projects	389	(1)
7.7 Bibliographical notes	390	(3)
8 Statistical Inference for Normal Populations	393	(50)
8.1 Introduction; parametric inference	393	(13)
8.2 Confidence intervals for one-sample model	406	(8)
8.3 From confidence intervals to hypothesis testing	414	(8)
8.4 Statistical inference for two-sample normal models	422	(5)
8.5 Regression analysis for the normal model	427	(8)
8.6 Testing for goodness-of-fit	435	(3)
8.7 Experiments, exercises, and projects	438	(2)
8.8 Bibliographical notes	440	(3)
9 Analysis of Variance	443	(18)
9.1 Single-factor ANOVA	443	(5)
9.2 Two-factor ANOVA	448	(8)
9.3 Experiments, exercises, and projects	456	(3)
9.4 Bibliographical notes	459	(2)
A Uncertainty Principle in Signal Processing and Quantum Mechanics	461	(4)
B Fuzzy Systems and Logic	465	(4)
C A Critique of Pure Reason	469	(4)
D The Remarkable Bernoulli Family	473	(4)
E Uncertain Virtual Worlds Mathematica Packages	477	(20)
F Tables	497	(6)
Index	503