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• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• 1. Introduction and Overview
• 1.1 Making decisions under uncertainty
• 1.2 Overview of this book
• Summary and conclusions
• Exercises
• Part I: Probability and Random Variables
• 2. Probability
• 2.1 Events and their probabilities
• 2.1.1 Outcomes, events, and the sample space
• 2.1.2 Set operations
• 2.2 Rules of Probability
• 2.2.1 Axioms of Probability
• 2.2.2 Computing probabilities of events
• 2.2.3 Applications in reliability
• 2.3 Combinatorics
• 2.3.1 Equally likely outcomes
• 2.3.2 Permutations and combinations
• 2.4 Conditional probability and independence
• Summary and conclusions
• Exercises
• 3. Discrete Random Variables and Their Distributions
• 3.1 Distribution of a random variable
• 3.1.1 Main concepts
• 3.1.2 Types of random variables
• 3.2 Distribution of a random vector
• 3.2.1 Joint distribution and marginal distributions
• 3.2.2 Independence of random variables
• 3.3 Expectation and variance
• 3.3.1 Expectation
• 3.3.2 Expectation of a function
• 3.3.3 Properties
• 3.3.4 Variance and standard deviation
• 3.3.5 Covariance and correlation
• 3.3.6 Properties
• 3.3.7 Chebyshev’s inequality
• 3.3.8 Application to finance
• 3.4 Families of discrete distributions
• 3.4.1 Bernoulli distribution
• 3.4.2 Binomial distribution
• 3.4.3 Geometric distribution
• 3.4.4 Negative Binomial distribution
• 3.4.5 Poisson distribution
• 3.4.6 Poisson approximation of Binomial distribution
• Summary and conclusions
• Exercises
• 4. Continuous Distributions
• 4.1 Probability density
• 4.2 Families of continuous distributions
• 4.2.1 Uniform distribution
• 4.2.2 Exponential distribution
• 4.2.3 Gamma distribution
• 4.2.4 Normal distribution
• 4.3 Central Limit Theorem
• Summary and conclusions
• Exercises
• 5. Computer Simulations and Monte Carlo Methods
• 5.1 Introduction
• 5.1.1 Applications and examples
• 5.2 Simulation of random variables
• 5.2.1 Random number generators
• 5.2.2 Discrete methods
• 5.2.3 Inverse transform method
• 5.2.4 Rejection method
• 5.2.5 Generation of random vectors
• 5.2.6 Special methods
• 5.3 Solving problems by Monte Carlo methods
• 5.3.1 Estimating probabilities
• 5.3.2 Estimating means and standard deviations
• 5.3.3 Forecasting
• 5.3.4 Estimating lengths, areas, and volumes
• 5.3.5 Monte Carlo integration
• Summary and conclusions
• Exercises
• Part II: Stochastic Processes
• 6. Stochastic Processes
• 6.1 Definitions and classifications
• 6.2 Markov processes and Markov chains
• 6.2.1 Markov chains
• 6.2.2 Matrix approach
• 6.2.3 Steady-state distribution
• 6.3 Counting processes
• 6.3.1 Binomial process
• 6.3.2 Poisson process
• 6.4 Simulation of stochastic processes
• Summary and conclusions
• Exercises
• 7. Queuing Systems
• 7.1 Main components of a queuing system
• 7.2 The Little’s Law
• 7.3 Bernoulli single-server queuing process
• 7.3.1 Systems with limited capacity
• 7.4 M/M/1 system
• 7.4.1 Evaluating the system’s performance
• 7.5 Multiserver queuing systems
• 7.5.1 Bernoulli k-server queuing process
• 7.5.2 M/M/k systems
• 7.5.3 Unlimited number of servers and M/M/∞
• 7.6 Simulation of queuing systems
• Summary and conclusions
• Exercises
• Part III: Statistics
• 8. Introduction to Statistics
• 8.1 Population and sample, parameters and statistics
• 8.2 Descriptive statistics
• 8.2.1 Mean
• 8.2.2 Median
• 8.2.3 Quantiles, percentiles, and quartiles
• 8.2.4 Variance and standard deviation
• 8.2.5 Standard errors of estimates
• 8.2.6 Interquartile range
• 8.3 Graphical statistics
• 8.3.1 Histogram
• 8.3.2 Stem-and-leaf plot
• 8.3.3 Boxplot
• 8.3.4 Scatter plots and time plots
• Summary and conclusions
• Exercises
• 9. Statistical Inference I
• 9.1 Parameter estimation
• 9.1.1 Method of moments
• 9.1.2 Method of maximum likelihood
• 9.1.3 Estimation of standard errors
• 9.2 Confidence intervals
• 9.2.1 Construction of confidence intervals: a general method
• 9.2.2 Confidence interval for the population mean
• 9.2.3 Confidence interval for the difference between two means
• 9.2.4 Selection of a sample size
• 9.2.5 Estimating means with a given precision
• 9.3 Unknown standard deviation
• 9.3.1 Large samples
• 9.3.2 Confidence intervals for proportions
• 9.3.3 Estimating proportions with a given precision
• 9.3.4 Small samples: Student’s t distribution
• 9.3.5 Comparison of two populations with unknown variances
• 9.4 Hypothesis testing
• 9.4.1 Hypothesis and alternative
• 9.4.2 Type I and Type II errors: level of significance
• 9.4.3 Level α tests: general approach
• 9.4.4 Rejection regions and power
• 9.4.5 Standard Normal null distribution (Z-test)
• 9.4.6 Z-tests for means and proportions
• 9.4.7 Pooled sample proportion
• 9.4.8 Unknown σ: T-tests
• 9.4.9 Duality: two-sided tests and two-sided confidence intervals
• 9.4.10 P-value
• 9.5 Inference about variances
• 9.5.1 Variance estimator and Chi-square distribution
• 9.5.2 Confidence interval for the population variance
• 9.5.3 Testing variance
• 9.5.4 Comparison of two variances. F-distribution
• 9.5.5 Confidence interval for the ratio of population variances
• 9.5.6 F-tests comparing two variances
• Summary and conclusions
• Exercises
• 10. Statistical Inference II
• 10.1 Chi-square tests
• 10.1.1 Testing a distribution
• 10.1.2 Testing a family of distributions
• 10.1.3 Testing independence
• 10.2 Nonparametric statistics
• 10.2.1 Sign test
• 10.2.2 Wilcoxon signed rank test
• 10.2.3 Mann–Whitney–Wilcoxon rank sum test
• 10.3 Bootstrap
• 10.3.1 Bootstrap distribution and all bootstrap samples
• 10.3.2 Computer generated bootstrap samples
• 10.3.3 Bootstrap confidence intervals
• 10.4 Bayesian inference
• 10.4.1 Prior and posterior
• 10.4.2 Bayesian estimation
• 10.4.3 Bayesian credible sets
• 10.4.4 Bayesian hypothesis testing
• Summary and conclusions
• Exercises
• 11. Regression
• 11.1 Least squares estimation
• 11.1.1 Examples
• 11.1.2 Method of least squares
• 11.1.3 Linear regression
• 11.1.4 Regression and correlation
• 11.1.5 Overfitting a model
• 11.2 Analysis of variance, prediction, and further inference
• 11.2.1 ANOVA and R-square
• 11.2.2 Tests and confidence intervals
• 11.2.3 Prediction
• 11.3 Multivariate regression
• 11.3.1 Introduction and examples
• 11.3.2 Matrix approach and least squares estimation
• 11.3.3 Analysis of variance, tests, and prediction
• 11.4 Model building
• 11.4.1 Adjusted R-square
• 11.4.2 Extra sum of squares, partial F-tests, and variable selection
• 11.4.3 Categorical predictors and dummy variables
• Summary and conclusions
• Exercises
• Appendix
• A.1 Data sets
• A.2 Inventory of distributions
• A.2.1 Discrete families
• A.2.2 Continuous families
• A.3 Distribution tables
• A.4 Calculus review
• A.4.1 Inverse function
• A.4.2 Limits and continuity
• A.4.3 Sequences and series
• A.4.4 Derivatives, minimum, and maximum
• A.4.5 Integrals
• A.5 Matrices and linear systems
• A.6 Answers to selected Exercises
• Index