CRC standard probability and statistics tables and formulae

Description

• Includes examples to illustrate important topics and their applications • Defines several realistic data sets to explain key concepts • Addresses fundamental topics, along with regression analysis, control theory, ANOVA, decision theory, and signal processing • Provides standardized test sets to illustrate algorithms Whether you are a statistician, engineer, or businessperson, you need statistics. You want to be able to easily reference tables, find formulas, and know how to use them so you can extract information from data without getting bogged down by advanced statistical methods. Your goal is to determine the appropriate statistical procedures and interpret the results. Standard Probability and Statistics: Tables and Formulae provides the tools you need to do just that. Logically organized and reaching far beyond a mere catalog, a textual description accompanies each entry - most include an example. The topics addressed are directly applicable to modern business and engineering as well as to statistics, including regression analysis, ANOVA, decision theory, signal processing, and control theory. The result is an accessible, example-oriented handbook that supplies the basic principles, the most commonly used values, and the information to make them work for you. It is easy to fill a statistics reference with hundreds of pages of tables - sometimes for just one test. This handbook is much more. With topics ranging from classical statistics to modern applications, Standard Probability and Statistics fills the need for an up-to-date, authoritative statistics reference. 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