Preface p. xi
Probability p. 1
Introduction p. 1
Sample Spaces p. 2
Probability Measures p. 4
Computing Probabilities: Counting Methods p. 6
The Multiplication Principle p. 7
Permutations and Combinations p. 9
Conditional Probability p. 16
Independence p. 23
Concluding Remarks p. 26
Problems p. 26
Random Variables p. 35
Discrete Random Variables p. 35
Bernoulli Random Variables p. 37
The Binomial Distribution p. 38
The Geometric and Negative Binomial Distributions p. 40
The Hypergeometric Distribution p. 42
The Poisson Distribution p. 42
Continuous Random Variables p. 47
The Exponential Density p. 50
The Gamma Density p. 53
The Normal Distribution p. 54
The Beta Density p. 58
Functions of a Random Variable p. 58
Concluding Remarks p. 64
Problems p. 64
Joint Distributions p. 71
Introduction p. 71
Discrete Random Variables p. 72
Continuous Random Variables p. 75
Independent Random Variables p. 84
Conditional Distributions p. 87
The Discrete Case p. 87
The Continuous Case p. 88
Functions of Jointly Distributed Random Variables p. 96
Sums and Quotients p. 96
The General Case p. 99
Extrema and Order Statistics p. 104
Problems p. 107
Expected Values p. 116
The Expected Value of a Random Variable p. 116
Expectations of Functions of Random Variables p. 121
Expectations of Linear Combinations of Random Variables p. 124
Variance and Standard Deviation p. 130
A Model for Measurement Error p. 135
Covariance and Correlation p. 138
Conditional Expectation and Prediction p. 147
Definitions and Examples p. 147
Prediction p. 152
The Moment-Generating Function p. 155
Approximate Methods p. 161
Problems p. 166
Limit Theorems p. 177
Introduction p. 177
The Law of Large Numbers p. 177
Convergence in Distribution and the Central Limit Theorem p. 181
Problems p. 188
Distributions Derived from the Normal Distribution p. 192
Introduction p. 192
x[superscript 2], t, and F Distributions p. 192
The Sample Mean and the Sample Variance p. 195
Problems p. 198
Survey Sampling p. 199
Introduction p. 199
Population Parameters p. 200
Simple Random Sampling p. 202
The Expectation and Variance of the Sample Mean p. 203
Estimation of the Population Variance p. 210
The Normal Approximation to the Sampling Distribution of X p. 214
Estimation of a Ratio p. 220
Stratified Random Sampling p. 227
Introduction and Notation p. 227
Properties of Stratified Estimates p. 228
Methods of Allocation p. 232
Concluding Remarks p. 238
Problems p. 239
Estimation of Parameters and Fitting of Probability Distributions p. 255
Introduction p. 255
Fitting the Poisson Distribution to Emissions of Alpha Particles p. 255
Parameter Estimation p. 257
The Method of Moments p. 260
The Method of Maximum Likelihood p. 267
Maximum Likelihood Estimates of Multinomial Cell Probabilities p. 272
Large Sample Theory for Maximum Likelihood Estimates p. 274
Confidence Intervals from Maximum Likelihood Estimates p. 279
The Bayesian Approach to Parameter Estimation p. 285
Further Remarks on Priors p. 294
Large Sample Normal Approximation to the Posterior p. 296
Computational Aspects p. 297
Efficiency and the Cramer-Rao Lower Bound p. 298
An Example: The Negative Binomial Distribution p. 302
Sufficiency p. 305
A Factorization Theorem p. 306
The Rao-Blackwell Theorem p. 310
Concluding Remarks p. 311
Problems p. 312
Testing Hypotheses and Assessing Goodness of Fit p. 329
Introduction p. 329
The Neyman-Pearson Paradigm p. 331
Specification of the Significance Level and the Concept of a p-value p. 334
The Null Hypothesis p. 335
Uniformly Most Powerful Tests p. 336
The Duality of Confidence Intervals and Hypothesis Tests p. 337
Generalized Likelihood Ratio Tests p. 339
Likelihood Ratio Tests for the Multinomial Distribution p. 341
The Poisson Dispersion Test p. 347
Hanging Rootograms p. 349
Probability Plots p. 352
Tests for Normality p. 358
Concluding Remarks p. 361
Problems p. 362
Summarizing Data p. 377
Introduction p. 377
Methods Based on the Cumulative Distribution Function p. 378
The Empirical Cumulative Distribution Function p. 378
The Survival Function p. 380
Quantile-Quantile Plots p. 385
Histograms, Density Curves, and Stem-and-Leaf Plots p. 389
Measures of Location p. 392
The Arithmetic Mean p. 393
The Median p. 395
The Trimmed Mean p. 397
M Estimates p. 397
Comparison of Location Estimates p. 398
Estimating Variability of Location Estimates by the Bootstrap p. 399
Measures of Dispersion p. 401
Boxplots p. 402
Exploring Relationships with Scatterplots p. 404
Concluding Remarks p. 407
Problems p. 408
Comparing Two Samples p. 420
Introduction p. 420
Comparing Two Independent Samples p. 421
Methods Based on the Normal Distribution p. 421
Power p. 433
A Nonparametric Method-The Mann-Whitney Test p. 435
Bayesian Approach p. 443
Comparing Paired Samples p. 444
Methods Based on the Normal Distribution p. 446
A Nonparametric Method-The Signed Rank Test p. 448
An Example-Measuring Mercury Levels in Fish p. 450
Experimental Design p. 452
Mammary Artery Ligation p. 452
The Placebo Effect p. 453
The Lanarkshire Milk Experiment p. 453
The Portacaval Shunt p. 454
FD&C Red No. 40 p. 455
Further Remarks on Randomization p. 456
Observational Studies, Confounding, and Bias in Graduate Admissions p. 457
Fishing Expeditions p. 458
Concluding Remarks p. 459
Problems p. 459
The Analysis of Variance p. 477
Introduction p. 477
The One-Way Layout p. 477
Normal Theory; the F Test p. 478
The Problem of Multiple Comparisons p. 485
A Nonparametric Method-The Kruskal-Wallis Test p. 488
The Two-Way Layout p. 489
Additive Parametrization p. 489
Normal Theory for the Two-Way Layout p. 492
Randomized Block Designs p. 500
A Nonparametric Method-Friedman's Test p. 503
Concluding Remarks p. 504
Problems p. 505
The Analysis of Categorical Data p. 514
Introduction p. 514
Fisher's Exact Test p. 514
The Chi-Square Test of Homogeneity p. 516
The Chi-Square Test of Independence p. 520
Matched-Pairs Designs p. 523
Odds Ratios p. 526
Concluding Remarks p. 530
Problems p. 530
Linear Least Squares p. 542
Introduction p. 542
Simple Linear Regression p. 547
Statistical Properties of the Estimated Slope and Intercept p. 547
Assessing the Fit p. 550
Correlation and Regression p. 560
The Matrix Approach to Linear Least Squares p. 564
Statistical Properties of Least Squares Estimates p. 567
Vector-Valued Random Variables p. 567
Mean and Covariance of Least Squares Estimates p. 573
Estimation of [gamma superscript 2] p. 575
Residuals and Standardized Residuals p. 576
Inference about [beta] p. 577
Multiple Linear Regression-An Example p. 580
Conditional Inference, Unconditional Inference, and the Bootstrap p. 585
Local Linear Smoothing p. 587
Concluding Remarks p. 591
Problems p. 591
Common Distributions p. A1
Tables p. A4
Bibliography p. A25
Answers to Selected Problems p. A32
Author Index p. A48
Applications Index p. A51
Subject Index p. A54