Bonamente M. Statistics and Analysis of Scientific Data

Springer, 2013. — 303 p. — ISBN: 1461479835, 9781461479833Statistics and Analysis of Scientific Data covers the foundations of probability theory and statistics, and a number of numerical and analytical methods that are essential for the present-day analyst of scientific data.
Topics covered include probability theory, distribution functions of statistics, fits to two-dimensional datasheets and parameter estimation, Monte Carlo methods and Markov chains. Equal attention is paid to the theory and its practical application, and results from classic experiments in various fields are used to illustrate the importance of statistics in the analysis of scientific data.
The main pedagogical method is a theory-then-application approach, where emphasis is placed first on a sound understanding of the underlying theory of a topic, which becomes the basis for an efficient and proactive use of the material for practical applications. The level is appropriate for undergraduates and beginning graduate students, and as a reference for the experienced researcher. Basic calculus is used in some of the derivations, and no previous background in probability and statistics is required. The book includes many numerical tables of data, as well as exercises and examples to aid the students' understanding of the topic.
Keywords » 2-variable dataset - Monte Carlo Markov chain methods - Probability and applied statistics textbook - data analysis methods - distribution functions of statistics - numerical Monte Carlo methods - probability textbook - random variable theories of probability - statistical applications physics - statistical data textbookTheory of Probability.- Random Variables and Their Distribution.- Sum and Functions of Random Variables.- Estimate of Mean and Variance and Confidence Intervals.- Distribution Function of Statistics and Hypothesis Testing.- Maximum Likelihood Fit to a Two-Variable Dataset.- Goodness of Fit and Parameter Uncertainty.- Comparison Between Models.- Monte Carlo Methods.- Markov Chains and Monte Carlo Markov Chains.- A: Numerical Tables.- B: Solutions.Theory of Probability
Experiments, Events and the Sample Space
Probability of Events
The Kolmogorov Axioms
Frequentist or Classical Method
Bayesian or Empirical Method
Fundamental Properties of Probability
Statistical Independence
Conditional Probability
The Total Probability Theorem and Bayes’ TheoremRandom Variables and Their Distribution
Random Variables
Probability Distribution Functions
Moments of a Distribution Function
The Mean and the Sample Mean
The Variance and the Sample Variance
Covariance and Correlation Between Random Variables
Joint Distribution and Moments of Two
Random Variables
Statistical Independence of Random Variables
Three Fundamental Distributions
The Binomial Distribution
The Gaussian Distribution
The Poisson Distribution
Comparison of Binomial, Gaussian and Poisson DistributionsSum and Functions of Random Variables
Linear Combination of Random Variables
General Mean and Variance Formulas
Uncorrelated Variables and the /
The Moment Generating Function
Properties of the Moment Generating Function
The Moment Generating Function of the
Gaussian and Poisson Distribution
The Central Limit Theorem
The Distribution of Functions of Random Variables
The Method of Change of Variables
A Method for Multi-dimensional Functions
The Law of Large Numbers
The Mean of Functions of Random Variables
The Variance of Functions of Random Variables and Error Propagation Formulas
Sum of a Constant
Weighted Sum of Two Variables
Product and Division of Two Random Variables
Power of a Random Variable
Exponential of a Random Variable
Logarithm of a Random Variable
The Quantile Function and Simulation of Random Variables
General Method to Simulate a Variable
Simulation of a Gaussian VariableEstimate of Mean and Variance and Confidence Intervals
The Maximum Likelihood Method for Gaussian Variables
Estimate of the Mean
Estimate of the Variance
Estimate of Mean for Non-uniform Uncertainties
Linear Average,Weighted Mean and Median
The Maximum Likelihood Method for Other Distributions
Method of Moments
Quantiles and Confidence Intervals
Confidence Interval for a Gaussian Variable
Confidence Intervals for the Mean of a Poisson Variable
Upper and Lower Limits
Limits for a Gaussian Variable
Limits for a Poisson Variable
Bayesian Methods for the Poisson Mean
Bayesian Upper and Lower Limits for a Poisson Variable
Bayesian Expectation of the Poisson MeanDistribution Function of Statistics and Hypothesis Testing
Statistics and Hypothesis Testing
The χ-2 Distribution
Derivation of the Probability Distribution Function
Moments and Other Properties of the χ Distribution
The Sampling Distribution of the Variance
The F Statistic
Derivation of the Probability Distribution Function
Moments and Other Properties of the F Distribution
The Sampling Distribution of the Mean and the Student’s t Distribution
Comparison of Sample Mean with Parent Mean
Comparison of Two Sample MeansMaximum Likelihood Fit to a Two-Variable Dataset
Measurement of Pairs of Variables
Maximum Likelihood Fit to a Gaussian Dataset
Least-Squares Fit to a Straight Line, or Linear Regression
Multiple Linear Regression
Best-Fit Parameters for Multiple Regression
Parameter Errors and Covariance Between Parameters
Errors and Covariance of Linear Regression Parameters
Special Cases: Identical Errors or No Errors Available
Maximum Likelihood Fit to a Non Linear Function
Linear Regression with Poisson DataGoodness of Fit and Parameter Uncertainty
Goodness of Fit for the χ Fit Statistic
Systematic Errors and Intrinsic Scatter
Goodness of Fit for the Cash C Statistic
Confidence Intervals on Model Parameters
Confidence Interval on All Parameters
Confidence Intervals on Reduced Number of Parameters
Confidence Intervals for Poisson Data
The Linear Correlation Coefficient
Fitting Two-Variable Datasets with Bi-variate Errors
Generalized Least-Squares Linear Fit to Bi-variate Data
Linear Fit Using Bi-variate Errors in the χ StatisticComparison Between Models
The F-Test
F-Test for Comparison of Two Independent χ-2 Measurements
F-Test for an Additional Model Component
Kolmogorov-Smirnov Tests
Comparison of Data to a Model
Two-Sample Kolmogorov-Smirnov TestMonte Carlo Methods
Traditional Monte Carlo Integration
Dart’ Method Monte Carlo Integration and Function Evaluation
Simulation of Random Variables
Monte Carlo Estimates of Errors for Two-Dimensional Datasets
The Bootstrap Method
The Jackknife MethodMarkov Chains and Monte Carlo Markov Chains
Stochastic Processes and Markov Chains
Mathematical Properties of Markov Chains
Recurrent and Transient States
Limiting Probabilities and Stationary Distribution
Implementation of Monte Carlo Markov Chains
Requirements and Goals of a Monte Carlo
Markov Chain
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Tests of Convergence
The Geweke Z Score Test
The Gelman-Rubin Test
The Raftery-Lewis TestA Numerical Tables
A The Gaussian Distribution
A The Error Function and Integrals of the Gaussian Distribution
A Upper and Lower Limits for a Poisson Distribution
A The χ-2 Distribution
A The F Distribution
A The Student’s t Distribution
A The Linear Correlation Coefficient r
A The Kolmogorov-Smirnov TestB Solutions
B Chapter
B Chapter
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B ChapterReferences
Index