Introductory Statistics and Random Phenomena

I Descriptive Statistics-Compressing Data.- 1 Why One Needs to Analyze Data.- 1.1 Coin tossing, lottery, and the stock market.- 1.2 Inventory problems in management.- 1.3 Battery life and quality control in manufacturing.- 1.4 Reliability of complex systems.- 1.5 Point processes in time and space.- 1.6 Polls-social sciences.- 1.7 Time series.- 1.8 Repeated experiments and testing.- 1.9 Simple chaotic dynamical systems.- 1.10 Complex dynamical systems.- 1.11 Pseudorandom number generators and the Monte-Carlo methods.- 1.12 Fractals and image reconstruction.- 1.13 Coding and decoding, unbreakable ciphers.- 1.14 Experiments, exercises, and projects.- 1.15 Bibliographical notes.- 2 Data Representation and Compression.- 2.1 Data types, categorical data.- 2.2 Numerical data: order statistics, median, quantiles.- 2.3 Numerical data: histograms, means, moments.- 2.4 Location, dispersion, and shape parameters.- 2.5 Probabilities: a frequentist viewpoint.- 2.6 Multidimensional data: histograms and other graphical representations.- 2.7 2-D data: regression and correlations.- 2.8 Fractal data.- 2.9 Measuring information content:entropy.- 2.10 Experiments, exercises, and projects.- 2.11 Bibliographical notes.- 3 Analytic Representation of Random Experimental Data.- 3.1 Repeated experiments and the law of large numbers.- 3.2 Characteristics of experiments: distribution functions, densities, means, variances.- 3.3 Uniform distributions, simulation of random quantities, the Monte Carlo method.- 3.4 Bernoulli and binomial distributions.- 3.5 Rescaling probabilities: Poisson approximation.- 3.6 Stability of Fluctuations Law: Gaussian approximation.- 3.7 How to estimate p in Bernoulli experiments.- 3.8 Other continuous distributions; Gamma function calculus.- 3.9 Testing the fit of a distribution.- 3.10 Random vectors and multivariate distributions.- 3.11 Experiments, exercises, and projects.- 3.12 Bibliographical notes.- II Modeling Uncertainty.- 4 Algorithmic Complexity and Random Strings.- 4.1 Heart of randomness: when is random - random?.- 4.2 Computable strings and the Turing machine.- 4.3 Kolmogorov complexity and random strings.- 4.4 Typical sequences: Martin-Lof tests of randomness.- 4.5 Stability of subsequences: von Mises randomness.- 4.6 Computable framework of randomness: degrees of irregularity.- 4.7 Experiments, exercises, and projects.- 4.8 Bibliographical notes.- 5 Statistical Independence and Kolmogorov's Probability Theory.- 5.1 Description of experiments, random variables, and Kolmogorov's axioms.- 5.2 Uniform discrete distributions and counting.- 5.3 Statistical independence as a model for repeated experiments..- 5.4 Expectations and other characteristics of random variables.- 5.4.1 Expectations.- 5.4.2 Expectations of functions of random variables. Variance.- 5.4.3 Expectations of functions of vectors. Covariance.- 5.4.4 Expectation of the product. Variance of the sum of independent random variables.- 5.4.5 Moments and the moment generating function.- 5.4.6 Expectations of general random variables.- 5.5 Averages of independent random variables.- 5.6 Laws of large numbers and small deviations.- 5.7 Central limit theorem and large deviations.- 5.8 Experiments, exercises, and projects.- 5.9 Bibliographical Notes.- 6 Chaos in Dynamical Systems: How Uncertainty Arises in Scientific and Engineering Phenomena.- 6.1 Dynamical systems: general concepts and typical examples.- 6.2 Orbits and fixed points.- 6.3 Stability of frequencies and the ergodic theorem.- 6.4 Stability of fluctuations and the central limit theorem.- 6.5 Attractors, fractals, and entropy.- 6.6 Experiments, exercises, and projects.- 6.7 Bibliographical notes.- III Model Specification-Design of Experiments.- 7 General Principles of Statistical Analysis.- 7.1 Design of experiments and planning of investigation.- 7.2 Model selection.- 7.3 Determining the method of statistical inference.- 7.3.1 Maximum likelihood estimator (MLE).- 7.3.2 Least squares estimator (LSE).- 7.3.3 Method of moments (MM).- 7.3.4 Concluding remarks.- 7.4 Estimation of fractal dimension.- 7.5 Practical side of data collection and analysis.- 7.6 Experiments, exercises, and projects.- 7.7 Bibliographical notes.- 8 Statistical Inference for Normal Populations.- 8.1 Introduction; parametric inference.- 8.2 Confidence intervals for one-sample model.- 8.3 From confidence intervals to hypothesis testing.- 8.4 Statistical inference for two-sample normal models.- 8.5 Regression analysis for the normal model.- 8.6 Testing for goodness-of-fit.- 8.7 Experiments, exercises, and projects.- 8.8 Bibliographical notes.- 9 Analysis of Variance.- 9.1 Single-factor ANOVA.- 9.2 Two-factor ANOVA.- 9.3 Experiments, exercises, and projects.- 9.4 Bibliographical notes.- A Uncertainty Principle in Signal Processing and Quantum Mechanics.- B Fuzzy Systems and Logic.- C A Critique of Pure Reason.- D The Remarkable Bernoulli Family.- F Tables.