Introduction to Probability and Statistics for Engineers and Scientists

Contents
Preface
Organization and coverage
Preface
Acknowledgments
1 Introduction to statistics
1.1 Introduction
1.2 Data collection and descriptive statistics
1.3 Inferential statistics and probability models
1.4 Populations and samples
1.5 A brief history of statistics
Problems
2 Descriptive statistics
2.1 Introduction
2.2 Describing data sets
2.2.1 Frequency tables and graphs
2.2.2 Relative frequency tables and graphs
2.2.3 Grouped data, histograms, ogives, and stem and leaf plots
2.3 Summarizing data sets
2.3.1 Sample mean, sample median, and sample mode
Germ-Free Mice
Conventional Mice
2.3.2 Sample variance and sample standard deviation
An algebraic identity
2.3.3 Sample percentiles and box plots
2.4 Chebyshev's inequality
Chebyshev's inequality
The one-sided Chebyshev inequality
2.5 Normal data sets
The empirical rule
2.6 Paired data sets and the sample correlation coefficient
Properties of r
2.7 The Lorenz curve and Gini index
2.8 Using R
Problems
3 Elements of probability
3.1 Introduction
3.2 Sample space and events
3.3 Venn diagrams and the algebra of events
3.4 Axioms of probability
3.5 Sample spaces having equally likely outcomes
Basic principle of counting
Proof of the Basic Principle
Notation and terminology
3.6 Conditional probability
3.7 Bayes' formula
3.8 Independent events
Problems
4 Random variables and expectation
4.1 Random variables
4.2 Types of random variables
4.3 Jointly distributed random variables
4.3.1 Independent random variables
4.3.2 Conditional distributions
4.4 Expectation
Remarks
4.5 Properties of the expected value
4.5.1 Expected value of sums of random variables
4.6 Variance
Remark
Remark
4.7 Covariance and variance of sums of random variables
4.8 Moment generating functions
4.9 Chebyshev's inequality and the weak law of large numbers
Problems
5 Special random variables
5.1 The Bernoulli and binomial random variables
5.1.1 Using R to calculate binomial probabilities
5.2 The Poisson random variable
5.2.1 Using R to calculate Poisson probabilities
5.3 The hypergeometric random variable
5.4 The uniform random variable
5.5 Normal random variables
5.6 Exponential random variables
5.6.1 The Poisson process
5.6.2 The Pareto distribution
5.7 The gamma distribution
5.8 Distributions arising from the normal
5.8.1 The chi-square distribution
5.8.1.1 The relation between chi-square and gamma random variables
5.8.2 The t-distribution
5.8.3 The F-distribution
5.9 The logistics distribution
5.10 Distributions in R
Problems
6 Distributions of sampling statistics
6.1 Introduction
6.2 The sample mean
6.3 The central limit theorem
6.3.1 Approximate distribution of the sample mean
6.3.2 How large a sample is needed?
6.4 The sample variance
6.5 Sampling distributions from a normal population
6.5.1 Distribution of the sample mean
6.5.2 Joint distribution of X and S2
6.6 Sampling from a finite population
Remark
Problems
7 Parameter estimation
7.1 Introduction
7.2 Maximum likelihood estimators
7.2.1 Estimating life distributions
7.3 Interval estimates
Remark
7.3.1 Confidence interval for a normal mean when the variance is unknown
Remarks
7.3.2 Prediction intervals
7.3.3 Confidence intervals for the variance of a normal distribution
7.4 Estimating the difference in means of two normal populations
Remark
7.5 Approximate confidence interval for the mean of a Bernoulli random variable
Remark
7.6 Confidence interval of the mean of the exponential distribution
7.7 Evaluating a point estimator
7.8 The Bayes estimator
Remark
Remark: On choosing a normal prior
Problems
8 Hypothesis testing
8.1 Introduction
8.2 Significance levels
8.3 Tests concerning the mean of a normal population
8.3.1 Case of known variance
Remark
8.3.1.1 One-sided tests
Remark
Remarks
8.3.2 Case of unknown variance: the t-test
8.4 Testing the equality of means of two normal populations
8.4.1 Case of known variances
8.4.2 Case of unknown variances
8.4.3 Case of unknown and unequal variances
8.4.4 The paired t-test
8.5 Hypothesis tests concerning the variance of a normal population
8.5.1 Testing for the equality of variances of two normal populations
8.6 Hypothesis tests in Bernoulli populations
8.6.1 Testing the equality of parameters in two Bernoulli populations
8.7 Tests concerning the mean of a Poisson distribution
8.7.1 Testing the relationship between two Poisson parameters
Problems
9 Regression
9.1 Introduction
9.2 Least squares estimators of the regression parameters
9.3 Distribution of the estimators
Remarks
Notation
Computational identity for SSR
9.4 Statistical inferences about the regression parameters
9.4.1 Inferences concerning β
Hypothesis test of H0: β= 0
Confidence interval for β
Remark
9.4.1.1 Regression to the mean
9.4.2 Inferences concerning α
Confidence interval estimator of α
9.4.3 Inferences concerning the mean response α+βx0
Confidence interval estimator of α+βx0
9.4.4 Prediction interval of a future response
Prediction interval for a response at the input level x0
Remarks
9.4.5 Summary of distributional results
9.5 The coefficient of determination and the sample correlation coefficient
9.6 Analysis of residuals: assessing the model
9.7 Transforming to linearity
9.8 Weighted least squares
Remarks
Remarks
9.9 Polynomial regression
Remark
9.10 Multiple linear regression
Remark
Remark
9.10.1 Predicting future responses
Confidence interval estimate of E [ Y|x] =∑ ki=0xiβi, (x 0≡ 1)
Prediction Interval for Y(x)
9.10.2 Dummy variables for categorical data
9.11 Logistic regression models for binary output data
Problems
10 Analysis of variance
10.1 Introduction
10.2 An overview
10.3 One-way analysis of variance
The sum of squares identity
10.3.1 Using R to do the computations
10.3.2 Multiple comparisons of sample means
10.3.3 One-way analysis of variance with unequal sample sizes
Remark
10.4 Two-factor analysis of variance: introduction and parameter estimation
10.5 Two-factor analysis of variance: testing hypotheses
10.6 Two-way analysis of variance with interaction
Problems
11 Goodness of fit tests and categorical data analysis
11.1 Introduction
11.2 Goodness of fit tests when all parameters are specified
Remarks
11.2.1 Determining the critical region by simulation
Remarks
11.3 Goodness of fit tests when some parameters are unspecified
11.4 Tests of independence in contingency tables
11.5 Tests of independence in contingency tables having fixed marginal totals
11.6 The Kolmogorov-Smirnov goodness of fit test for continuous data
Problems
12 Nonparametric hypothesis tests
12.1 Introduction
12.2 The sign test
12.3 The signed rank test
12.4 The two-sample problem
12.4.1 Testing the equality of multiple probability distributions
12.5 The runs test for randomness
Problems
13 Quality control
13.1 Introduction
13.2 Control charts for average values: the x control chart
Remarks
13.2.1 Case of unknown μ and σ
Technical remark
Remarks
13.3 S-control charts
13.4 Control charts for the fraction defective
Remark
13.5 Control charts for number of defects
13.6 Other control charts for detecting changes in the population mean
13.6.1 Moving-average control charts
13.6.2 Exponentially weighted moving-average control charts
13.6.3 Cumulative sum control charts
Problems
14 Life testing
14.1 Introduction
14.2 Hazard rate functions
Remark on terminology
14.3 The exponential distribution in life testing
14.3.1 Simultaneous testing - stopping at the rth failure
Remark
14.3.2 Sequential testing
14.3.3 Simultaneous testing - stopping by a fixed time
Remark
14.3.4 The Bayesian approach
Remark
14.4 A two-sample problem
14.5 The Weibull distribution in life testing
14.5.1 Parameter estimation by least squares
Remarks
Problems
15 Simulation, bootstrap statistical methods, and permutation tests
15.1 Introduction
15.2 Random numbers
15.2.1 The Monte Carlo simulation approach
15.3 The bootstrap method
15.4 Permutation tests
15.4.1 Normal approximations in permutation tests
15.4.2 Two-sample permutation tests
15.5 Generating discrete random variables
15.6 Generating continuous random variables
15.6.1 Generating a normal random variable
15.7 Determining the number of simulation runs in a Monte Carlo study
Problems
16 Machine learning and big data
16.1 Introduction
16.2 Late flight probabilities
16.3 The naive Bayes approach
16.3.1 A variation of naive Bayes approach
16.4 Distance-based estimators. The k-nearest neighbors rule
16.4.1 A distance-weighted method
16.4.2 Component-weighted distances
16.5 Assessing the approaches
16.6 When characterizing vectors are quantitative
16.6.1 Nearest neighbor rules
16.6.2 Logistics regression
16.7 Choosing the best probability: a bandit problem
Remarks
Problems
Appendix of Tables
Index