CRC standard probability and statistics tables and formulae

Preface......Page 4
Acknowlegments......Page 5
Contents......Page 6
1.2 DATA SETS......Page 12
1.3 REFERENCES......Page 13
CHAPTER 2: Summarizing Data......Page 14
2.1.2 Frequency distribution......Page 15
2.1.4 Frequency polygons......Page 16
2.1.5 Chernoff faces......Page 18
2.2 NUMERICAL SUMMARY MEASURES......Page 19
2.2.1 (Arithmetic) mean......Page 20
2.2.4 Harmonic mean......Page 21
2.2.7 p % trimmed mean......Page 22
2.2.8 Quartiles......Page 23
2.2.12 Variance......Page 24
2.2.14.1 Standard error of the mean......Page 25
2.2.19 Box plots......Page 26
2.2.23 Moments......Page 28
2.2.25 Measures of kurtosis......Page 29
2.2.27 Sheppard’s corrections for grouping......Page 30
CHAPTER 3: Probability......Page 31
3.1 ALGEBRA OF SETS......Page 32
3.2.1 The product rule for ordered pairs......Page 34
3.2.5 Combinations (binomial coefficients)......Page 35
3.2.6 Sample selection......Page 36
3.2.8 Multinomial coefficients......Page 37
3.3 PROBABILITY......Page 38
3.3.4 Probability theorems......Page 39
3.3.6 Conditional probability......Page 40
3.3.8 The law of total probability......Page 41
3.3.10 Independence......Page 42
3.4.1.2 Cumulative distribution function......Page 43
3.5.1 Expected value......Page 44
3.5.3.1 Moments about the origin......Page 45
3.5.4.1 Moment generating function......Page 46
3.5.4.2 Factorial moment generating functions......Page 47
3.5.4.4 Cumulant generating function......Page 48
3.6.2 Continuous case......Page 49
3.6.4 Moments......Page 50
3.6.5 Marginal distributions......Page 51
3.6.7 Conditional distributions......Page 52
3.6.8 Variance and covariance......Page 53
3.6.10 Moment generating function......Page 54
3.6.12.1 Joint probability distribution......Page 55
3.6.12.2 Cumulative distribution function......Page 56
3.6.12.4 Conditional distributions......Page 57
3.7 INEQUALITIES......Page 58
CHAPTER 4: Functions of Random Variables......Page 60
4.1.1 Method of distribution functions......Page 61
4.1.2 Method of transformations (one variable)......Page 62
4.1.3 Method of transformations (two or more variables)......Page 63
4.1.4 Method of moment generating functions......Page 64
4.2.1 Deterministic sums of random variables......Page 65
4.3.1 Definitions......Page 66
4.3.4 The law of large numbers......Page 67
4.4 FINITE POPULATION......Page 68
4.5.2 Theorems: the t distribution......Page 69
4.6.1 Definition......Page 70
4.6.5 Joint distributions......Page 71
4.6.7 Uniform distribution: order statistics......Page 72
4.6.8.1 Expected value of normal order statistics......Page 73
4.6.8.2 Variances and covariances of order statistics......Page 75
4.7.1 Probability integral of the range......Page 77
4.7.2 Percentage points, studentized range......Page 85
CHAPTER 5: Discrete Probability Distributions......Page 89
5.1.1 Properties......Page 90
5.2.1 Properties......Page 91
5.4.1 Properties......Page 92
5.4.3 Tables......Page 93
5.5.3 Tables......Page 100
5.6.2 Variates......Page 102
5.6.3 Tables......Page 103
5.8.1 Properties......Page 109
5.8.2 Variates......Page 110
5.9 POISSON DISTRIBUTION......Page 111
5.9.3 Tables......Page 112
5.10.1 Properties......Page 118
CHAPTER 6: Continuous Probability Distributions......Page 119
6.1.2 Probability density function......Page 122
6.2.1 Properties......Page 123
6.2.3 Related distributions......Page 124
6.3.3 Related distributions......Page 125
6.4.1 Properties......Page 126
6.4.3 Related distributions......Page 127
6.4.4 Critical values for chi–square distribution......Page 128
6.5.1 Properties......Page 133
6.6.2 Probability density function......Page 134
6.6.3 Related distributions......Page 135
6.7.3 Related distributions......Page 136
6.8.1 Properties......Page 137
6.8.3 Related distributions......Page 138
6.8.4 Critical values for the F distribution......Page 139
6.9.3 Related distributions......Page 146
6.10.2 Probability density function......Page 147
6.11.3 Related distributions......Page 148
6.12.3 Related distributions......Page 149
6.13.2 Probability density function......Page 150
6.14.1 Properties......Page 151
6.14.3 Related distributions......Page 152
6.15.2 Probability density function......Page 153
6.16.3 Related distributions......Page 154
6.17.2 Probability density function......Page 155
6.18.2 Probability density function......Page 156
6.18.3 Related distributions......Page 157
6.19.2 Probability density function......Page 158
6.20.3 Related distributions......Page 159
6.21.2 Probability density function......Page 160
6.21.3 Related distributions......Page 161
6.22.2 Probability density function......Page 162
6.23.3 Related distributions......Page 163
6.23.4 Critical values for the t distribution......Page 164
6.24.2 Probability density function......Page 166
6.25.3 Related distributions......Page 167
6.26.3 Related distributions......Page 168
6.27.1 Other relationships among distributions......Page 169
7.1 THE PROBABILITY DENSITY FUNCTION AND RELATED FUNCTIONS......Page 173
7.3 TOLERANCE FACTORS FOR NORMAL DISTRIBUTIONS......Page 183
7.3.1 Tables of tolerance intervals for normal distributions......Page 185
7.4.2 Two-sample Z test......Page 186
7.5 MULTIVARIATE NORMAL DISTRIBUTION......Page 189
7.6 DISTRIBUTION OF THE CORRELATION COEFFICIENT FOR A BIVARIATE NORMAL......Page 190
7.6.2 Zero correlation coefficient for bivariate normal......Page 191
7.7 CIRCULAR NORMAL PROBABILITIES......Page 193
7.8 CIRCULAR ERROR PROBABILITIES......Page 194
8.1 DEFINITIONS......Page 195
8.2 CRAMER–RAO INEQUALITY......Page 196
8.3 THEOREMS......Page 197
8.5 THE LIKELIHOOD FUNCTION......Page 198
8.8 DIFFERENT ESTIMATORS......Page 199
8.9 ESTIMATORS FOR MEAN AND STANDARD DEVIATION IN SMALL SAMPLES......Page 201
8.10 ESTIMATORS FOR MEAN AND STANDARD DEVIATION IN LARGE SAMPLES......Page 202
9.1 DEFINITIONS......Page 203
9.3 SAMPLE SIZE CALCULATIONS......Page 204
9.5.1 Confidence interval for mean of normal population,known variance......Page 206
9.5.2 Confidence interval for mean of normal population, unknown variance......Page 207
9.5.3 Confidence interval for variance of normal population......Page 208
9.5.6.1 Table of confidence interval for medians......Page 209
9.5.7 Confidence interval for parameter in a Poisson distribution......Page 210
9.5.8 Confidence interval for parameter in a binomial distribution......Page 213
9.6.3 Confidence interval for difference in means, unequal unknown variances......Page 230
9.6.6 Difference in success probabilities......Page 231
9.6.7 Difference in medians......Page 232
9.7 FINITE POPULATION CORRECTION FACTOR......Page 233
10.1 INTRODUCTION......Page 234
10.1.1 Tables......Page 235
10.2 THE NEYMAN–PEARSON LEMMA......Page 238
10.4 GOODNESS OF FIT TEST......Page 239
10.5 CONTINGENCY TABLES......Page 240
10.6 BARTLETT’S TEST......Page 241
10.6.2 Tables for Bartlett’s test......Page 242
10.7.1 Tables for Cochran’s test......Page 245
10.8 NUMBER OF OBSERVATIONS REQUIRED FOR THE COMPARISON OF A POPULATION VARIANCE WITH A STANDARD VALUE USING THE CHI–SQUARE......Page 248
10.9 CRITICAL VALUES FOR TESTING OUTLIERS......Page 249
10.10 TEST OF SIGNIFICANCE IN 2x2 CONTINGENCY TABLES......Page 251
10.11 DETERMINING VALUES IN BERNOULLI TRIALS......Page 267
11.1 SIMPLE LINEAR REGRESSION......Page 268
11.1.2 Sum of squares......Page 270
11.1.4 The mean response......Page 271
11.1.5 Prediction interval......Page 272
11.1.8 Sample correlation coefficient......Page 273
11.1.9 Example......Page 274
11.2 MULTIPLE LINEAR REGRESSION......Page 275
11.2.2 Sum of squares......Page 276
11.2.4 The mean response......Page 277
11.2.6 Analysis of variance table......Page 278
11.2.8 Partial F test......Page 279
11.2.9 Residual analysis......Page 280
11.2.10 Example......Page 281
11.3 ORTHOGONAL POLYNOMIALS......Page 282
11.3.1 Tables for orthogonal polynomials......Page 285
CHAPTER 12: Analysis of Variance......Page 288
12.1.1 Sum of squares......Page 289
12.1.3 Analysis of variance table......Page 290
12.1.4.2 Duncan’s multiple range test......Page 291
12.1.4.3 Duncan’s multiple range test......Page 292
12.1.4.5 Tables for Dunnett’s procedure......Page 296
12.1.5 Contrasts......Page 298
12.1.6 Example......Page 299
12.2.1.2 Sum of squares......Page 301
12.2.2 Analysis of variance table......Page 302
12.2.3.1 Models and assumptions......Page 303
12.2.3.3 Mean squares and properties......Page 304
12.2.3.4 Analysis of variance table......Page 305
12.2.4.1 Models and assumptions......Page 306
12.2.4.2 Sum of squares......Page 307
12.2.4.3 Mean squares and properties......Page 308
12.2.4.4 Analysis of variance table......Page 309
12.2.5.1 Models and assumptions......Page 310
12.2.5.3 Mean squares and properties......Page 311
12.2.5.4 Analysis of variance table......Page 312
12.2.6 Example......Page 313
12.3.1 Models and assumptions......Page 315
12.3.2 Sum of squares......Page 316
12.3.3 Mean squares and properties......Page 318
12.3.4 Analysis of variance table......Page 321
12.5 FACTOR ANALYSIS......Page 324
12.6.1 Models and assumptions......Page 326
12.6.3 Mean squares and properties......Page 327
12.6.4 Analysis of variance table......Page 328
13.1 LATIN SQUARES......Page 330
13.2 GRAECO–LATIN SQUARES......Page 331
13.3 BLOCK DESIGNS......Page 332
13.4 FACTORIAL EXPERIMENTATION: 2 FACTORS......Page 334
13.5 2 r FACTORIAL EXPERIMENTS......Page 335
13.7 TABLES FOR DESIGN OF EXPERIMENTS......Page 337
13.7.1 Plans of factorial experiments confounded in randomized incomplete blocks......Page 338
13.7.2 Plans of 2 n factorials in fractional replication......Page 341
13.7.3 Plans of incomplete block designs......Page 343
13.7.4 Main effect and interactions in factorial designs......Page 345
13.8 REFERENCES......Page 348
CHAPTER 14: Nonparametric Statistics......Page 349
14.2 KENDALL’S RANK CORRELATION COEFFICIENT......Page 350
14.2.1 Tables for Kendall rank correlation coefficient......Page 351
14.3.2 Two-sample Kolmogorov–Smirnoff test......Page 352
14.3.3.1 Critical values, one-sample Kolmogorov–Smirnoff test......Page 354
14.3.3.2 Critical values, two-sample Kolmogorov–Smirnoff test......Page 355
14.4 KRUSKAL–WALLIS TEST......Page 357
14.4.1 Tables for Kruskal–Wallis test......Page 358
14.5 THE RUNS TEST......Page 359
14.5.1 Tables for the runs test......Page 360
14.6.1 Table of critical values for the sign test......Page 371
14.7 SPEARMAN’S RANK CORRELATION COEFFICIENT......Page 372
14.7.1 Tables for Spearman’s rank correlation coefficient......Page 373
14.8 WILCOXON MATCHED-PAIRS SIGNED-RANKS TEST......Page 377
14.9 WILCOXON RANK –SUM (MANN –WHITNEY)TEST......Page 378
14.9.1 Tables for Wilcoxon (Mann –Whitney) U statistic......Page 379
14.9.2 Critical values for Wilcoxon (Mann –Whitney) statistic......Page 383
14.10 WILCOXON SIGNED-RANK TEST......Page 387
15.1.1 Control charts......Page 389
15.2 ACCEPTANCE SAMPLING......Page 392
15.2.1 Sequential sampling......Page 393
15.2.1.1 Sequential probability ratio tests......Page 395
15.2.1.3 One-sided variance test......Page 396
15.3.1 Failure time distributions......Page 397
15.4 RISK ANALYSIS AND DECISION RULES......Page 398
CHAPTER 16: General Linear Models......Page 402
16.2.1 The simple linear regression model......Page 403
16.2.2 Multiple linear regression......Page 405
16.2.3 One-way analysis of variance......Page 406
16.2.4 Two-way analysis of variance......Page 407
16.2.5 Analysis of covariance......Page 408
16.3.2 Determinants and partitioning of determinants......Page 409
16.3.4 Eigenvalues......Page 410
16.3.5.2 Properties......Page 411
16.3.6 Additional definitions and properties......Page 412
16.4.1 Multivariate distributions......Page 413
16.4.3 Minimum variance unbiased estimates......Page 414
16.5.2 Simple linear regression......Page 415
16.5.3 Analysis of variance, one-way anova......Page 416
16.5.4 Multiple linear regression......Page 417
16.5.5 Randomized blocks (one observation per cell)......Page 418
16.5.6 Quadratic form due to hypothesis......Page 419
16.5.7 Sum of squares due to error......Page 420
16.5.9 Computation procedure for hypothesis testing......Page 421
16.6 GENERAL LINEAR MODEL OF LESS THAN FULL RANK......Page 422
16.6.1 Estimable function and estimability......Page 423
16.6.2.1 Sum of squares due to error......Page 425
16.6.2.2 Sum of squares due to hypothesis......Page 426
16.6.3 Constraints and conditions......Page 427
CHAPTER 17: Miscellaneous Topics......Page 428
17.1 GEOMETRIC PROBABILITY......Page 430
17.2.1 Discrete entropy......Page 432
17.2.3 Channel capacity......Page 434
17.2.4 Shannon\'s theorem......Page 435
17.3 KALMAN FILTERING......Page 436
17.4.1 Theory......Page 437
17.4.3 Example: Insurance company......Page 438
17.5.1 Transition function......Page 439
17.5.3 Recurrence......Page 440
17.5.4 Stationary distributions......Page 441
17.5.6 Ehrenfest chain......Page 442
17.7 MEASURE THEORETICAL PROBABILITY......Page 443
17.8 MONTE CARLO INTEGRATION TECHNIQUES......Page 444
17.8.2 Hit-or-miss Monte Carlo method......Page 445
17.9 QUEUING THEORY......Page 446
17.9.3 M/M/2 queue......Page 448
17.9.6 M/M/c/K queue......Page 449
17.9.9 M/D/1 queue......Page 450
17.10 RANDOM MATRIX EIGENVALUES......Page 451
17.10.1.1 Vibonacci numbers......Page 454
17.11.1.1 Linear congruential generators......Page 455
17.11.1.3 Lagged-Fibonacci generators......Page 456
17.11.2 Generating nonuniform random variables......Page 457
17.11.2.1 Discrete random variables......Page 459
17.12 RESAMPLING METHODS......Page 460
17.13.1 Definitions......Page 461
17.14.1 Estimation......Page 462
17.14.4 Mean filter......Page 463
17.15.1 Brownian motion (Wiener processes)......Page 464
17.15.2 Brownian motion expectations......Page 465
17.15.4 Stochastic integration......Page 467
17.15.5 Stochastic differential equations......Page 468
17.15.6 Motion in a domain......Page 469
17.15.7 Option Pricing......Page 470
17.16.3 Bertrand’s box “paradox ”......Page 471
17.16.5 Bingo cards: nontransitive......Page 472
17.16.8.1 Shuffling cards......Page 473
17.16.8.2 Card games......Page 474
17.16.9.1 Even odds from a biased coin......Page 475
17.16.10 Coupon collectors problem......Page 476
17.16.11.3 Dice: same distribution......Page 477
17.16.14 Gambler ’s ruin problem......Page 478
17.16.15 Gender distributions......Page 479
17.16.17.2 Pooling of blood samples......Page 480
17.16.18.1 Ratio of uniform numbers......Page 481
17.16.19 Lotteries......Page 482
17.16.23 Multi-armed bandit problem......Page 483
17.16.25 Passage problems......Page 484
17.16.28.1 Long runs......Page 485
17.16.28.3 Waiting times: many types of characters......Page 486
17.16.28.4 First random sequence......Page 487
17.16.29.2 Random walk in two dimensions (Rayleigh problem)......Page 488
17.16.29.3 Random walk in three dimensions......Page 489
17.16.31 Roots of a random polynomial......Page 490
17.16.33 Simpson paradox......Page 491
17.17.1 Statlib......Page 492
17.17.2 Uniform resource locators......Page 493
17.17.3 Interactive demonstrations and tutorials......Page 494
17.17.4 Online textbooks, reference manuals, and journals......Page 496
17.17.5 Free statistical software packages......Page 498
17.17.6 Demonstration statistical software packages......Page 501
17.18.1 Random deviates......Page 502
17.18.3 Combinations......Page 505
CHAPTER 18: Special Functions......Page 509
18.1.1 Differential equation......Page 510
18.1.3 Recurrence relations......Page 511
18.1.5 Integral representations......Page 512
18.1.9 Modified Bessel functions......Page 513
18.1.9.1 Relation to ordinary Bessel functions......Page 514
18.2.2 Properties......Page 515
18.5 ERROR FUNCTIONS......Page 516
18.6.1 Exponentiation......Page 517
18.6.4 Circular functions and exponentials......Page 518
18.8 GAMMA FUNCTION......Page 519
18.8.1 Other integrals for the gamma function......Page 520
18.8.2 Properties......Page 521
18.8.5 Digamma function......Page 522
18.8.6 Incomplete gamma functions......Page 523
18.9.2.1 Special cases......Page 524
18.10.2 Special values......Page 525
18.10.7 Derivative and integration formulae......Page 526
18.13.1 Stirling numbers......Page 527
18.14 SUMS OF POWERS OF INTEGERS......Page 529
18.15 TABLES OF ORTHOGONAL POLYNOMIALS......Page 531
18.16 REFERENCES......Page 532
List of Notation......Page 533