This book presents in a compact form the program carried out in introductory statistics courses and discusses some essential topics for research activity, such as Monte Carlo simulation techniques, methods of statistical inference, best fit and analysis of laboratory data. All themes are developed starting from fundamentals, highlighting their applicative aspects, up to the detailed description of several cases particularly relevant for technical and scientific research. The text is dedicated to university students in scientific fields and to all researchers who have to solve practical problems by applying data analysis and simulation procedures. The R software is adopted throughout the book, with a rich library of original programs accessible to the readers through a website.
Author(s): Alberto Rotondi, Paolo Pedroni, Antonio Pievatolo
Series: UNITEXT, 139
Publisher: Springer
Year: 2022
Language: English
Pages: 642
City: Cham
Preface
How to Use the Text
Contents
About the Authors
1 Probability
1.1 Chance, Chaos and Determinism
1.2 Some Basic Terms
1.3 The Concept of Probability
1.4 Axiomatic Probability
1.5 Repeated Trials
1.6 Elements of Combinatorial Analysis
1.7 Bayes' Theorem
1.8 Learning Algorithms
1.9 Problems
2 Representation of Random Phenomena
2.1 Introduction
2.2 Random Variables
2.3 Cumulative or Distribution Function
2.4 Data Representation
2.5 Discrete Random Variables
2.6 Binomial Distribution
2.7 Continuous Random Variables
2.8 Mean, Sum of Squares, Variance, Standard Deviation and Quantiles
2.9 Operators
2.10 Simple Random Sample
2.11 Convergence Criteria
2.12 Problems
3 Basic Probability Theory
3.1 Introduction
3.2 Properties of the Binomial Distribution
3.3 Poisson Distribution
3.4 Normal or Gaussian Density
3.5 The Three-Sigma Law and the Standard Gaussian Density
3.6 Central Limit Theorem and Universality of the GaussianCurve
3.7 Poisson Stochastic Processes
3.8 χ2 Density
3.9 Uniform Density
3.10 Chebyshev's Inequality
3.11 How to Use Probability Calculus
3.12 Problems
4 Multivariate Probability Theory
4.1 Introduction
4.2 Multivariate Statistical Distributions
4.3 Covariance and Correlation
4.4 Two-Dimensional Gaussian Distribution
4.5 The General Multidimensional Case
4.6 Multivariate Probability Regions
4.7 Multinomial Distribution
4.8 Problems
5 Functions of Random Variables
5.1 Introduction
5.2 Functions of a Random Variable
5.3 Functions of Several Random Variables
5.4 Mean and Variance Transformation
5.5 Means and Variances for n Variables
5.6 Problems
6 Basic Statistics: Parameter Estimation
6.1 Introduction
6.2 Confidence Intervals
6.3 Confidence Intervals with Pivotal Variables
6.4 Mention of the Bayesian Approach
6.5 Some Notations
6.6 Probability Estimation
6.7 Probability Estimation from Large Samples
6.8 Poissonian Interval Estimation
6.9 Mean Estimation from Large Samples
6.10 Variance Estimation from Large Samples
6.11 Mean and Variance Estimation for Gaussian Samples
6.12 How to Use the Estimation Theory
6.13 Estimates from a Finite Population
6.14 Histogram Analysis
6.15 Estimation of the Correlation
6.16 Problems
7 Basic Statistics: Hypothesis Testing
7.1 Testing One Hypothesis
7.2 The Gaussian z-Test
7.3 Student's t-Test
7.4 Chi-Square Test
7.5 Compatibility Check Between Sample and Population
7.6 Hypothesis Testing with Contingency Tables
7.7 Multiple Tests
7.8 Snedecor's F-Test
7.9 Analysis of Variance (ANOVA)
7.10 Two-Way ANOVA
7.11 Problems
8 Monte Carlo Methods
8.1 Introduction
8.2 What Is Monte Carlo?
8.3 Mathematical Aspects
8.4 Generation of Discrete Random Variables
8.5 Generation of Continuous Random Variables
8.6 Linear Search Method
8.7 Rejection Method
8.8 Particular Random Generation Methods
8.9 Monte Carlo Analysis of Distributions
8.10 Evaluation of Confidence Intervals
8.11 Simulation of Counting Experiments
8.12 Non-parametric Bootstrap
8.13 Hypothesis Test with Simulated Data
8.14 Problems
9 Applications of Monte Carlo Methods
9.1 Introduction
9.2 Study of Diffusion Phenomena
9.3 Simulation of Stochastic Processes
9.4 Number of Workers in a Plant: Synchronous Simulation
9.5 Number of Workers in a Plant: Asynchronous Simulation
9.6 Kolmogorov-Smirnov Test
9.7 Metropolis Algorithm
9.8 Ising Model
9.9 Definite Integral Calculation
9.10 Importance Sampling
9.11 Stratified Sampling
9.12 Multidimensional Integrals
9.13 Problems
10 Statistical Inference and Likelihood
10.1 Introduction
10.2 Maximum Likelihood (ML) Method
10.3 Estimator Properties
10.4 Theorems on Estimators
10.5 Confidence Intervals
10.6 Least Squares Method and Maximum Likelihood
10.7 Best Fit of Densities to Data and Histograms
10.8 Weighted Mean
10.9 Test of Hypotheses
10.10 One- or Two-Sample Tests
10.11 Most Powerful Tests
10.12 Test Functions
10.13 Sequential Tests
10.14 Problems
11 Least Squares
11.1 Introduction
11.2 No Errors on Predictors
11.3 Errors in Predictors
11.4 Least Squares Regression Lines: Unweighted Case
11.5 Unweighted Linear Least Squares
11.6 Weighted Linear Least Squares
11.7 Properties of Least Squares Estimates
11.8 Model Testing and Search for Functional Forms
11.9 Search for Correlations
11.10 Fit Strategies
11.11 Nonlinear Least Squares
11.12 Problems
12 Experimental Data Analysis
12.1 Introduction
12.2 Terminology
12.3 Constant and Variable Physical Quantities
12.4 Instrumental Sensitivity and Accuracy
12.5 Measurement Uncertainty
12.6 Treatment of Systematic Effects
12.7 Best Fit with Offset Systematic Errors
12.8 Best Fit with Scale Systematic Errors
12.9 Indirect Measurements and Error Propagation
12.10 Measurement Types
12.11 M(0, 0, Δ) Measurements
12.12 M(0, σ, 0) Measurements
12.13 M(0, σ, Δ) Measurements
12.14 M(f, 0, 0) Measurements
12.15 M(f, σ, 0), M(f, 0, Δ) and M(f, σ, Δ) Measurements
12.16 A Case Study: Millikan's Experiments
12.17 Some Remarks on the Scientific Method
12.18 Problems
A Table of Symbols
B R Software
C Moment-Generating Functions
D Solutions of Problems
E Tables
E.1 Integral of the Gaussian Density
E.2 Quantiles of the Student's Density
E.3 Integrals of the Reduced χ2 Density
E.4 Quantile Values of the Non-Reduced χ2 Density
E.5 Quantiles of the F Density
Bibliography
Index
