Mathematical statistics with applications

Front Cover......Page 1
Title Page......Page 6
Copyright......Page 7
Contents......Page 8
Preface......Page 16
Note to the Student......Page 24
1.1 Introduction......Page 26
1.2 Characterizing a Set of Measurements: Graphical Methods......Page 28
Exercises......Page 31
1.3 Characterizing a Set of Measurements: Numerical Methods......Page 33
Exercises......Page 36
1.4 How Inferences Are Made......Page 38
1.5 Theory and Reality......Page 39
References and Further Readings......Page 40
Supplementary Exercises......Page 41
2.1 Introduction......Page 45
2.2 Probability and Inference......Page 46
2.3 A Review of Set Notation......Page 48
Exercises......Page 50
2.4 A Probabilistic Model for an Experiment: The Discrete Case......Page 51
Exercises......Page 57
2.5 Calculating the Probability of an Event: The Sample-Point Method......Page 60
Exercises......Page 64
2.6 Tools for Counting Sample Points......Page 65
Exercises......Page 73
2.7 Conditional Probability and the Independence of Events......Page 76
Exercises......Page 79
2.8 Two Laws of Probability......Page 82
Exercises......Page 84
2.9 Calculating the Probability of an Event: The Event-Composition Method......Page 87
Exercises......Page 93
2.10 The Law of Total Probability and Bayes’ Rule......Page 95
Exercises......Page 97
2.11 Numerical Events and Random Variables......Page 100
2.12 Random Sampling......Page 102
2.13 Summary......Page 104
Supplementary Exercises......Page 105
3.1 Basic Definition......Page 111
3.2 The Probability Distribution for a Discrete Random Variable......Page 112
Exercises......Page 115
3.3 The Expected Value of a Random Variable or a Function of a Random Variable......Page 116
Exercises......Page 122
3.4 The Binomial Probability Distribution......Page 125
Exercises......Page 135
3.5 The Geometric Probability Distribution......Page 139
Exercises......Page 144
3.6 The Negative Binomial Probability Distribution (Optional)......Page 146
Exercises......Page 148
3.7 The Hypergeometric Probability Distribution......Page 150
Exercises......Page 153
3.8 The Poisson Probability Distribution......Page 156
Exercises......Page 161
3.9 Moments and Moment-Generating Functions......Page 163
Exercises......Page 167
3.10 Probability-Generating Functions (Optional)......Page 168
3.11 Tchebysheff’s Theorem......Page 171
Exercises......Page 172
3.12 Summary......Page 174
References and Further Readings......Page 175
Supplementary Exercises......Page 176
4.1 Introduction......Page 182
4.2 The Probability Distribution for a Continuous Random Variable......Page 183
Exercises......Page 191
4.3 Expected Values for Continuous Random Variables......Page 195
Exercises......Page 197
4.4 The Uniform Probability Distribution......Page 199
Exercises......Page 201
4.5 The Normal Probability Distribution......Page 203
Exercises......Page 206
4.6 The Gamma Probability Distribution......Page 210
Exercises......Page 214
4.7 The Beta Probability Distribution......Page 219
Exercises......Page 222
4.8 Some General Comment......Page 226
4.9 Other Expected Values......Page 227
Exercises......Page 231
4.10 Tchebysheff’s Theorem......Page 232
Exercises......Page 234
4.11 Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional)......Page 235
Exercises......Page 238
4.12 Summary......Page 239
Supplementary Exercises......Page 240
5.1 Introduction......Page 248
5.2 Bivariate and Multivariate Probability Distributions......Page 249
Exercises......Page 257
5.3 Marginal and Conditional Probability Distributions......Page 260
Exercises......Page 267
5.4 Independent Random Variables......Page 272
Exercises......Page 276
5.5 The Expected Value of a Function of Random Variables......Page 280
5.6 Special Theorems......Page 283
Exercises......Page 286
5.7 The Covariance of Two Random Variables......Page 289
Exercises......Page 293
5.8 The Expected Value and Variance of Linear Functions of Random Variables......Page 295
Exercises......Page 301
5.9 The Multinomial Probability Distribution......Page 304
Exercises......Page 307
5.10 The Bivariate Normal Distribution (Optional)......Page 308
Exercises......Page 309
5.11 Conditional Expectations......Page 310
Exercises......Page 314
5.12 Summary......Page 315
Supplementary Exercises......Page 316
6.1 Introduction......Page 321
6.2 Finding the Probability Distribution of a Function of Random Variables......Page 322
6.3 The Method of Distribution Functions......Page 323
Exercises......Page 332
6.4 The Method of Transformations......Page 335
Exercises......Page 341
6.5 The Method of Moment-Generating Functions......Page 343
Exercises......Page 347
6.6 Multivariable Transformations Using Jacobians (Optional)......Page 350
Exercises......Page 355
6.7 Order Statistics......Page 358
Exercises......Page 363
Supplementary Exercises......Page 366
7.1 Introduction......Page 371
Exercises......Page 374
7.2 Sampling Distributions Related to the Normal Distribution......Page 378
Exercises......Page 389
7.3 The Central Limit Theorem......Page 395
Exercises......Page 398
7.4 A Proof of the Central Limit Theorem (Optional)......Page 402
7.5 The Normal Approximation to the Binomial Distribution......Page 403
Exercises......Page 407
7.6 Summary......Page 410
Supplementary Exercises......Page 411
8.1 Introduction......Page 415
8.2 The Bias and Mean Square Error of Point Estimators......Page 417
Exercises......Page 419
8.3 Some Common Unbiased Point Estimators......Page 421
8.4 Evaluating the Goodness of a Point Estimator......Page 424
Exercises......Page 427
8.5 Confidence Intervals......Page 431
Exercises......Page 434
8.6 Large-Sample Confidence Intervals......Page 436
Exercises......Page 440
8.7 Selecting the Sample Size......Page 446
Exercises......Page 449
8.8 Small-Sample Confidence Intervals for μ and μ[sub(1)] – μ[sub(2)]......Page 450
Exercises......Page 455
8.9 Confidence Intervals for σ[sup(2)]......Page 459
Exercises......Page 461
References and Further Readings......Page 462
Supplementary Exercises......Page 463
9.1 Introduction......Page 469
9.2 Relative Efficiency......Page 470
Exercises......Page 472
9.3 Consistency......Page 473
Exercises......Page 479
9.4 Sufficiency......Page 484
Exercises......Page 487
9.5 The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation......Page 489
Exercises......Page 495
9.6 The Method of Moments......Page 497
Exercises......Page 500
9.7 The Method of Maximum Likelihood......Page 501
Exercises......Page 506
9.8 Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional)......Page 508
References and Further Readings......Page 510
Supplementary Exercises......Page 511
10.1 Introduction......Page 513
10.2 Elements of a Statistical Test......Page 514
Exercises......Page 519
10.3 Common Large-Sample Tests......Page 521
Exercises......Page 526
10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests......Page 532
Exercises......Page 535
10.5 Relationships Between Hypothesis-Testing Procedures and Confidence Intervals......Page 536
Exercises......Page 537
10.6 Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-Values......Page 538
Exercises......Page 541
10.7 Some Comments on the Theory of Hypothesis Testing......Page 543
10.8 Small-Sample Hypothesis Testing for μ and μ[sub(1)] – μ[sub(2)]......Page 545
Exercises......Page 551
10.9 Testing Hypotheses Concerning Variances......Page 555
Exercises......Page 562
10.10 Power of Tests and the Neyman–Pearson Lemma......Page 565
Exercises......Page 571
10.11 Likelihood Ratio Tests......Page 574
Exercises......Page 579
10.12 Summary......Page 581
Supplementary Exercises......Page 582
11 Linear Models and Estimation by Least Squares......Page 588
11.1 Introduction......Page 589
11.2 Linear Statistical Models......Page 591
11.3 The Method of Least Squares......Page 594
Exercises......Page 597
11.4 Properties of the Least-Squares Estimators: Simple Linear Regression......Page 602
Exercises......Page 607
11.5 Inferences Concerning the Parameters β[sub(i)]......Page 609
Exercises......Page 611
11.6 Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression......Page 614
Exercises......Page 617
11.7 Predicting a Particular Value of Y by Using Simple Linear Regression......Page 618
Exercises......Page 622
11.8 Correlation......Page 623
Exercises......Page 627
11.9 Some Practical Examples......Page 629
Exercises......Page 633
11.10 Fitting the Linear Model by Using Matrices......Page 634
Exercises......Page 639
11.11 Linear Functions of the Model Parameters: Multiple Linear Regression......Page 640
11.12 Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression......Page 641
Exercises......Page 646
11.13 Predicting a Particular Value of Y by Using Multiple Regression......Page 647
11.14 A Test for H[sub(0)]: β[sub(g+1)] = β[sub(g+2) = · · · = β[sub(k)] = 0......Page 649
Exercises......Page 655
11.15 Summary and Concluding Remarks......Page 658
Supplementary Exercises......Page 659
12.1 The Elements Affecting the Information in a Sample......Page 665
12.2 Designing Experiments to Increase Accuracy......Page 666
12.3 The Matched-Pairs Experiment......Page 669
Exercises......Page 673
12.4 Some Elementary Experimental Designs......Page 676
Exercises......Page 681
Supplementary Exercises......Page 682
13.1 Introduction......Page 686
13.2 The Analysis of Variance Procedure......Page 687
13.3 Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout......Page 692
13.4 An Analysis of Variance Table for a One-Way Layout......Page 696
Exercises......Page 697
13.5 A Statistical Model for the One-Way Layout......Page 702
13.6 Proof of Additivity of the Sums of Squares and E (MST) for a One-Way Layout (Optional)......Page 704
13.7 Estimation in the One-Way Layout......Page 706
Exercises......Page 708
13.8 A Statistical Model for the Randomized Block Design......Page 711
Exercises......Page 712
13.9 The Analysis of Variance for a Randomized Block Design......Page 713
Exercises......Page 716
13.10 Estimation in the Randomized Block Design......Page 720
13.11 Selecting the Sample Size......Page 721
13.12 Simultaneous Confidence Intervals for More Than One Parameter......Page 723
Exercises......Page 725
13.13 Analysis of Variance Using Linear Models......Page 726
13.14 Summary......Page 730
Supplementary Exercises......Page 731
14.1 A Description of the Experiment......Page 738
14.2 The Chi-Square Test......Page 739
14.3 A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test......Page 741
Exercises......Page 743
14.4 Contingency Tables......Page 746
Exercises......Page 750
14.5 r × c Tables with Fixed Row or Column Totals......Page 754
Exercises......Page 756
14.6 Other Applications......Page 759
References and Further Readings......Page 761
Supplementary Exercises......Page 762
15.1 Introduction......Page 766
15.2 A General Two-Sample Shift Model......Page 767
15.3 The Sign Test for a Matched-Pairs Experiment......Page 769
Exercises......Page 772
15.4 The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment......Page 775
Exercises......Page 778
15.5 Using Ranks for Comparing Two Population Distributions: Independent Random Samples......Page 780
15.6 The Mann–Whitney U Test: Independent Random Samples......Page 783
Exercises......Page 788
15.7 The Kruskal–Wallis Test for the One-Way Layout......Page 790
Exercises......Page 793
15.8 The Friedman Test for Randomized Block Designs......Page 796
Exercises......Page 799
15.9 The Runs Test: A Test for Randomness......Page 802
Exercises......Page 807
15.10 Rank Correlation Coefficient......Page 808
Exercises......Page 812
15.11 Some General Comments on Nonparametric Statistical Tests......Page 814
References and Further Readings......Page 815
Supplementary Exercises......Page 816
16.1 Introduction......Page 821
16.2 Bayesian Priors, Posteriors, and Estimators......Page 822
Exercises......Page 830
16.3 Bayesian Credible Intervals......Page 833
Exercises......Page 836
16.4 Bayesian Tests of Hypotheses......Page 838
Exercises......Page 840
16.5 Summary and Additional Comments......Page 841
References and Further Readings......Page 844
A1.1 Matrices and Matrix Algebra......Page 846
A1.2 Addition of Matrices......Page 847
A1.4 Matrix Multiplication......Page 848
A1.5 Identity Elements......Page 850
A1.6 The Inverse of a Matrix......Page 852
A1.8 A Matrix Expression for a System of Simultaneous Linear Equations......Page 853
A1.9 Inverting a Matrix......Page 855
A1.10 Solving a System of Simultaneous Linear Equations......Page 859
A1.11 Other Useful Mathematical Results......Page 860
Table 1 Discrete Distributions......Page 862
Table 2 Continuous Distributions......Page 863
Table 1 Binomial Probabilities......Page 864
Table 2 Table of e[sup(-x)]......Page 867
Table 3 Poisson Probabilities......Page 868
Table 4 Normal Curve Areas......Page 873
Table 5 Percentage Points of the t Distributions......Page 874
Table 6 Percentage Points of the χ2 Distributions......Page 875
Table 7 Percentage Points of the F Distributions......Page 877
Table 8 Distribution Function of U......Page 887
Table 9 Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50......Page 893
Table 10 Distribution of the Total Number of Runs R in Samples of Size (n1, n2); P(R < a)......Page 895
Table 11 Critical Values of Spearman’s Rank Correlation Coefficient......Page 897
Table 12 Random Numbers......Page 898
Answers to Exercises......Page 902
Index......Page 921