Probability and Statistics The Science of Uncertainty

by Evans, Michael J.; Rosenthal, Jeffrey S.

• ISBN13: 9780716747420
• ISBN10: 0716747421
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-07-25
• Publisher: W. H. Freeman

Summary

Unlike traditional introductory math/stat textbooks,Probability and Statistics: The Science of Uncertaintybrings a modern flavor based on incorporating the computer to the course and an integrated approach to inference. From the start the book integrates simulations into its theoretical coverage, and emphasizes the use of computer-powered computation throughout.* Math and science majors with just one year of calculus can use this text and experience a refreshing blend of applications and theory that goes beyond merely mastering the technicalities. They'll get a thorough grounding in probability theory, and go beyond that to the theory of statistical inference and its applications. An integrated approach to inference is presented that includes the frequency approach as well as Bayesian methodology. Bayesian inference is developed as a logical extension of likelihood methods. A separate chapter is devoted to the important topic of model checking and this is applied in the context of the standard applied statistical techniques. Examples of data analyses using real-world data are presented throughout the text. A final chapter introduces a number of the most important stochastic process models using elementary methods. *Note:An appendix in the book contains Minitab code for more involved computations. The code can be used by students as templates for their own calculations. If a software package like Minitab is used with the course then no programming is required by the students.

Table of Contents

Preface	ix
Probability Models	1	(32)
Probability: A Measure of Uncertainty	1	(3)
Why Do We Need Probability Theory?	2	(2)
Probability Models	4	(6)
Venn Diagrams and Subsets	7	(3)
Properties of Probability Models	10	(3)
Uniform Probability on Finite Spaces	13	(6)
Combinatorial Principles	14	(5)
Conditional Probability and Independence	19	(9)
Conditional Probability	20	(3)
Independence of Events	23	(5)
Continuity of P	28	(2)
Further Proofs (Advanced)	30	(3)
Random Variables and Distributions	33	(90)
Random Variables	33	(4)
Distributions of Random Variables	37	(3)
Discrete Distributions	40	(10)
Important Discrete Distributions	41	(9)
Continuous Distributions	50	(11)
Important Absolutely Continuous Distributions	52	(9)
Cumulative Distribution Functions	61	(11)
Properties of Distribution Functions	62	(1)
Cdfs of Discrete Distributions	62	(2)
Cdfs of Absolutely Continuous Distributions	64	(2)
Mixture Distributions	66	(3)
Distributions Neither Discrete Nor Continuous	69	(3)
One-Dimensional Change of Variable	72	(5)
The Discrete Case	72	(1)
The Continuous Case	73	(4)
Joint Distributions	77	(12)
Joint Cumulative Distribution Functions	77	(2)
Marginal Distributions	79	(1)
Joint Probability Functions	80	(2)
Joint Density Functions	82	(7)
Conditioning and Independence	89	(15)
Conditioning on Discrete Random Variables	90	(1)
Conditionig on Continuous Random Variables	91	(2)
Independence of Random Variables	93	(6)
Order Statistics	99	(5)
Multidimensional Change of Variable	104	(7)
The Discrete Case	104	(1)
The Continuous Case (Advanced)	105	(3)
Convolution	108	(3)
Simulating Probability Distributions	111	(8)
Simulating Discrete Distributions	112	(2)
Simulating Continuous Distributions	114	(5)
Further Proofs (Advanced)	119	(4)
Expectation	123	(66)
The Discrete Case	123	(12)
The Absolutely Continuous Case	135	(7)
Variance, Covariance, and Correlation	142	(12)
Generating Functions	154	(12)
Characteristic Functions (Advanced)	161	(5)
Conditional Expectation	166	(10)
Discrete Case	166	(2)
Absolutely Continuous Case	168	(1)
Double Expectations	169	(2)
Conditional Variance (Advanced)	171	(5)
Inequalities	176	(6)
Jensen's Inequality (Advanced)	179	(3)
General Expectations (Advanced)	182	(3)
Further Proofs (Advanced)	185	(4)
Sampling Distributions and Limits	189	(50)
Sampling Distributions	190	(3)
Convergence in Probability	193	(5)
The Weak Law of Large Numbers	195	(3)
Convergence with Probability 1	198	(4)
The Strong Law of Large Numbers	200	(2)
Convergence in Distribution	202	(11)
The Central Limit Theorem	204	(5)
The Central Limit Theorem and Assessing Error	209	(4)
Monte Carlo Approximations	213	(9)
Normal Distribution Theory	222	(9)
The Chi-Squared Distribution	223	(3)
The t Distribution	226	(1)
The F Distribution	227	(4)
Further Proofs (Advanced)	231	(8)
Statistical Inference	239	(42)
Why Do We Need Statistics?	239	(5)
Inference Using a Probability Model	244	(3)
Statistical Models	247	(7)
Data Collection	254	(12)
Finite Populations	254	(2)
Simple Random Sampling	256	(3)
Histograms	259	(2)
Survey Sampling	261	(5)
Some Basic Inferences	266	(15)
Descriptive Statistics	267	(4)
Plotting Data	271	(2)
Types of Inference	273	(8)
Likelihood Inference	281	(70)
The Likelihood Function	281	(10)
Sufficient Statistics	286	(5)
Maximum Likelihood Estimation	291	(11)
The Multidimensional Case (Advanced)	299	(3)
Inferences Based on the MLE	302	(27)
Standard Errors and Bias	303	(4)
Confidence Intervals	307	(6)
Testing Hypotheses and P-Values	313	(7)
Sample Size Calculations: Confidence Intervals	320	(2)
Sample Size Calculations: Power	322	(7)
Distribution-Free Methods	329	(13)
Method of Moments	330	(1)
Bootstrapping	331	(4)
The Sign Statistic and Inferences about Quantiles	335	(7)
Large Sample Behavior of the MLE (Advanced)	342	(9)
Bayesian Inference	351	(54)
The Prior and Posterior Distributions	352	(9)
Inferences Based on the Posterior	361	(22)
Estimation	364	(4)
Credible Intervals	368	(3)
Hypothesis Testing and Bayes Factors	371	(6)
Prediction	377	(6)
Bayesian Computations	383	(14)
Asymptotic Normality of the Posterior	383	(1)
Sampling from the Posterior	383	(6)
Sampling from the Posterior Using Gibbs Sampling (Advanced)	389	(8)
Choosing Priors	397	(5)
Further Proofs (Advanced)	402	(3)
Optimal Inferences	405	(44)
Optimal Unbiased Estimation	405	(13)
The Cramer-Rao Inequality (Advanced)	412	(6)
Optimal Hypothesis Testing	418	(12)
Likelihood Ratio Tests (Advanced)	426	(4)
Optimal Bayesian Inferences	430	(4)
Decision Theory (Advanced)	434	(10)
Further Proofs (Advanced)	444	(5)
Model Checking	449	(30)
Checking the Sampling Model	449	(22)
Residual and Probability Plots	456	(4)
The Chi-Squared Goodness of Fit Test	460	(5)
Prediction and Cross-Validation	465	(1)
What Do We Do When a Model Fails?	466	(5)
Checking the Bayesian Model	471	(6)
The Problem with Multiple Checks	477	(2)
Relationships Among Variables	479	(100)
Related Variables	480	(14)
Cause-Effect Relationships and Experiments	483	(3)
Design of Experiments	486	(8)
Categorical Response and Predictors	494	(11)
Random Predictor	494	(3)
Deterministic Predictor	497	(2)
Bayesian Formulation	499	(6)
Quantitative Response and Predictors	505	(38)
The Method of Least Squares	505	(2)
The Simple Linear Regression Model	507	(14)
Bayesian Simple Linear Model (Advanced)	521	(4)
The Multiple Linear Regression Model (Advanced)	525	(18)
Quantitative Response and Categorical Predictors	543	(25)
One Categorical Predictor (One-Way ANOVA)	543	(6)
Repeated Measures (Paired Comparisons)	549	(3)
Two Categorical Predictors (Two-Way ANOVA)	552	(7)
Randomized Blocks	559	(1)
One Categorical and One Quantitative Predictor	560	(8)
Categorical Response and Quantitative Predictors	568	(4)
Further Proofs (Advanced)	572	(7)
Advanced Topic --- Stochastic Processes	579	(60)
Simple Random Walk	579	(7)
The Distribution of the Fortune	580	(2)
The Gambler's Ruin Problem	582	(4)
Markov Chains	586	(18)
Examples of Markov Chains	587	(3)
Computing with Markov Chains	590	(3)
Stationary Distributions	593	(4)
Markov Chain Limit Theorem	597	(7)
Markov Chain Monte Carlo	604	(9)
The Metropolis-Hastings Algorithm	607	(3)
The Gibbs Sampler	610	(3)
Martingales	613	(7)
Definition of a Martingale	613	(2)
Expected Values	615	(1)
Stopping Times	616	(4)
Brownian Motion	620	(9)
Faster and Faster Random Walks	621	(1)
Brownian Motion as a Limit	622	(3)
Diffusions and Stock Prices	625	(4)
Poisson Processes	629	(2)
Further Proofs	631	(8)
Appendices	639	(1)
A Mathematical Background	639	(8)
A.1 Derivatives	639	(1)
A.2 Integrals	640	(1)
A.3 Infinite Series	641	(1)
A.4 Matrix Multiplication	642	(1)
A.5 Partial Derivatives	642	(1)
A.6 Multivariable Integrals	643	(4)
A.6.1 Nonrectangular Regions	644	(3)
B Computations	647	(6)
C Common Distributions	653	(4)
C.1 Discrete Distributions	653	(1)
C.2 Absolutely Continuous Distributions	654	(3)
D Tables	657	(20)
D.1 Random Numbers	658	(2)
D.2 Standard Normal Cdf	660	(1)
D.3 Chi-Squared Distribution Quantiles	661	(1)
D.4 t Distribution Quantiles	662	(1)
D.5 F Distribution Quantiles	663	(9)
D.6 Binomial Distribution Probabilities	672	(5)
Index	677