Handbook of statistical distributions with applications

Description

In the area of applied statistics, scientists use statistical distributions to model a wide range of practical problems, from modeling the size grade distribution of onions to modeling global positioning data. To apply these probability models successfully, practitioners and researchers must have a thorough understanding of the theory as well as a familiarity with the practical situations. The Handbook of Statistical Distributions with Applications is the first reference to combine popular probability distribution models, formulas, applications, and software to assist you in computing probabilities, percentiles, moments, and other statistics. Presenting both common and specialized probability distribution models as well as providing applications with practical examples, this handbook offers comprehensive coverage of plots of probability density functions, methods of computing probability and percentiles, algorithms for random number generation, and inference, including point estimation, hypothesis tests, and sample size determination. The book discusses specialized distributions, some nonparametric distributions, tolerance factors for a multivariate normal distribution, and the distribution of the sample correlation coefficient, among others. Developed by the author, the StatCal software, along with the text, offers a useful reference for computing various table values. By using the software, you can compute probabilities, parameters, and moments; find exact tests; and obtain exact confidence intervals for distributions, such as binomial, hypergeometric, Poisson, negative binomial, normal, lognormal, inverse Gaussian, and correlation coefficient. In the applied statistics world, the Handbook of Statistical Distributions with Applications is now the reference for examining distribution functions - including univariate, bivariate normal, and multivariate - their definitions, their use in statistical inference, and their algorithms for random number generation. Handbook of Statistical Distributions with Applications......Page 1 STATISTICS: Textbooks and Monographs......Page 4 Preface......Page 6 Contents......Page 8 0.1 Introduction......Page 21 0.2 Contents of StatCalc......Page 24 1.1 Random Variables and Expectations......Page 28 1.2.2 Moments......Page 31 1.2.3 Measures of Variability......Page 32 1.2.5 Other Measures......Page 33 1.3 Some Functions Relevant to Reliability......Page 34 1.4 Model Fitting......Page 35 1.4.2 The Chi-Square Goodness-of-Fit Test......Page 36 1.5.1 Moment Estimation......Page 37 Some Terminologies......Page 38 Errors and Powers......Page 39 Tolerance Intervals......Page 42 The Accept/Reject Method......Page 43 1.8 Some Special Functions......Page 44 2.1 Description......Page 47 2.2 Moments......Page 48 3.1 Description......Page 49 3.2 Moments......Page 50 3.3 Computing Table Values......Page 52 Illustrative Examples......Page 53 3.4.2 Power of the Exact Test......Page 54 3.5.1 An Exact Confidence Interval......Page 56 3.5.2 Computing Exact Limits and Sample Size Calculation......Page 57 3.6.1 An Unconditional Test......Page 58 3.6.2 Power of the Unconditional Test......Page 59 3.7 Fisher's Exact Test......Page 60 3.7.1 Calculation of p-Values......Page 61 3.7.2 Exact Powers......Page 62 3.8.2 Relation to Other Distributions......Page 63 3.9 Random Number Generation......Page 64 3.10 Computation of Probabilities......Page 66 4.1 Description......Page 69 4.2 Moments......Page 70 4.3 Computing Table Values......Page 72 Illustrative Examples......Page 73 4.4 Point Estimation......Page 74 4.5.1 An Exact Test......Page 75 4.5.2 Power of the Exact Test......Page 76 4.6.1 Confidence Intervals......Page 77 Expected Length......Page 78 4.7.1 The Test......Page 80 4.7.2 Power Calculation......Page 81 4.8.3 Approximations......Page 82 4.9 Random Number Generation......Page 83 4.10 Computation of Probabilities......Page 84 5.1 Description......Page 88 5.2 Moments......Page 89 5.3 Computing Table Values......Page 91 5.5.1 An Exact Test......Page 92 5.5.2 Powers of the Exact Test......Page 93 5.6.1 An Exact Confidence Interval......Page 94 5.7.1 A Conditional Test......Page 95 5.7.2 Powers of the Conditional Test......Page 97 5.9 A Test for the Difference between Two Means......Page 98 5.9.1 An Unconditional Test......Page 99 5.9.2 Powers of the Unconditional Test......Page 100 5.10 Model Fitting with Examples......Page 101 5.11.2 Relation to Other Distributions......Page 103 5.12 Random Number Generation......Page 104 5.13 Computation of Probabilities......Page 105 6.1 Description......Page 109 6.3 Computing Table Values......Page 110 6.4 Properties and Results......Page 111 6.5 Random Number Generation......Page 112 7.1 Description......Page 113 7.2 Moments......Page 114 Illustrative Examples......Page 116 7.5 A Test for the Proportion......Page 117 7.7.1 Properties......Page 119 7.8 Random Number Generation......Page 120 7.9 A Computational Method for Probabilities......Page 122 8.1 Description......Page 123 8.3 Computing Table Values......Page 125 8.4.2 Interval Estimation......Page 128 8.6 Random Number Generation......Page 129 8.7 A Computational Algorithm for Probabilities......Page 130 9.1 Description......Page 131 9.3 Inferences......Page 132 9.5 Random Number Generation......Page 133 10.1 Description......Page 134 An Example for Model Checking......Page 135 10.3 Computing Table Values......Page 138 Some Illustrative Examples......Page 140 10.4.1 Point Estimation......Page 142 Power Computation......Page 143 10.4.3 Interval Estimation for the Mean......Page 145 Some Illustrative Examples......Page 146 Test about a Normal Variance......Page 147 Confidence Interval for a Normal Variance......Page 148 10.5 Two-Sample Inference......Page 149 Interval Estimation for the Ratio of Variances......Page 150 10.5.2 Inference for the Difference between Two Means when the Variances are Equal......Page 151 Power of the Two-Sample t-test......Page 152 Interval Estimation of u1 -- u2......Page 153 10.5.3 Inference for the Difference between Two Means......Page 155 10.6.1 Two-Sided Tolerance Intervals......Page 157 10.6.2 One-Sided Tolerance Limits......Page 158 Applications......Page 159 10.6.3 Equal-Tail Tolerance Intervals......Page 160 10.6.4 Simultaneous Hypothesis Testing for Quantiles......Page 161 10.6.5 Tolerance Limits for One-Way Random E®ects Model......Page 162 10.7 Properties and Results......Page 164 10.8 Relation to Other Distributions......Page 165 10.9 Random Number Generation......Page 166 10.10 Computing the Distribution Function......Page 168 11.1 Description......Page 170 11.2 Moments......Page 171 11.4 Applications......Page 172 11.5.1 Properties......Page 173 11.5.2 Relation to Other Distributions......Page 174 11.5.3 Approximations......Page 175 11.7 Computing the Distribution Function......Page 176 12.1 Description......Page 177 12.3 Computing Table Values......Page 179 12.4.2 Relation to Other Distributions......Page 180 12.4.3 Series Expansions......Page 181 12.5 Random Number Generation......Page 182 12.6 A Computational Method for Probabilities......Page 183 13.1 Description......Page 184 13.2 Moments......Page 185 13.4 Distribution of the Maximum of Several /t/ Variables......Page 186 Simultaneous Con¯dence Intervals for the Treatment Means......Page 187 13.4.3 An Example......Page 188 13.5.2 Relation to Other Distributions......Page 189 13.5.3 Series Expansions for Cumulative Probability......Page 190 13.7 A Computational Method for Probabilities......Page 191 14.1 Description......Page 192 14.3 Computing Table Values......Page 193 Con¯dence Intervals......Page 194 14.5.2 Relation to Other Distributions......Page 195 14.6 Random Number Generation......Page 196 15.1 Description......Page 197 15.2 Moments......Page 198 15.3 Computing Table Values......Page 199 15.4 Applications with Some Examples......Page 200 15.5.1 Maximum Likelihood Estimators......Page 201 15.5.3 Interval Estimation......Page 202 15.6 Properties and Results......Page 203 15.7 Random Number Generation......Page 204 15.8 A Computational Method for Probabilities......Page 205 16.1 Description......Page 207 16.2 Moments......Page 208 16.3 Computing Table Values......Page 209 16.5 Applications with an Example......Page 210 16.6.1 An Identity and Recurrence Relations......Page 213 16.6.2 Relation to Other Distributions......Page 214 16.7 Random Number Generation......Page 215 16.8 Evaluating the Distribution Function......Page 217 17.1 Description......Page 219 17.3 Computing Table Values......Page 221 17.4 Applications......Page 222 17.5.3 Approximations to Percentiles......Page 223 17.7 Evaluating the Distribution Function......Page 224 18.1 Description......Page 228 18.4 Applications......Page 230 18.5.1 Properties......Page 231 18.6 Random Number Generation......Page 232 18.7 Evaluating the Distribution Function......Page 233 19.1 Description......Page 236 19.2 Moments......Page 237 19.4 Applications......Page 238 19.5.1 Properties......Page 239 19.7 Evaluating the Distribution Function......Page 240 20.1 Description......Page 244 20.2 Moments......Page 245 20.4.1 Maximum Likelihood Estimators......Page 246 20.5 Applications......Page 247 20.6 Relation to Other Distributions......Page 249 20.7 Random Number Generation......Page 250 21.1 Description......Page 251 21.2 Moments......Page 252 21.3 Computing Table Values......Page 253 21.5 Applications......Page 254 21.7 Random Number Generation......Page 255 22.1 Description......Page 256 22.2 Moments......Page 257 22.3 Computing Table Values......Page 258 22.5 Confidence Interval and Test for the Mean......Page 259 22.6 Inferences for the Difference between Two Means......Page 260 Illustrative Examples......Page 261 22.7 Inferences for the Ratio of Two Means......Page 262 22.9 Properties and Results......Page 263 22.11 Computation of Probabilities and Percentiles......Page 264 23.1 Description......Page 266 23.2 Moments......Page 267 23.4 Inferences......Page 268 23.5 Applications......Page 269 23.8 Computation of Probabilities and Percentiles......Page 270 24.1 Description......Page 271 24.2 Moments......Page 272 24.4 Applications......Page 273 24.5 Point Estimation......Page 274 24.8 Computation of Probabilities and Percentiles......Page 275 25.1 Description......Page 276 25.2 Moments......Page 277 25.4 Maximum Likelihood Estimators......Page 278 25.5 Applications......Page 279 25.8 Computation of Probabilities and Percentiles......Page 280 26.1 Description......Page 281 26.3 Computing Table Values......Page 282 26.4.1 Estimation Based on Sample Quantiles......Page 283 26.6 Properties and Results......Page 284 26.8 Computation of Probabilities and Percentiles......Page 285 27.1 Description......Page 286 27.2 Moments......Page 287 27.4 One-Sample Inference......Page 288 27.4.2 Confidence Interval for the Mean......Page 289 27.5.1 Inferences for the Di®erence between Two Mean......Page 290 27.6 Random Number Generation......Page 292 27.7 Computational Methods for Probabilities and Percentiles......Page 293 28.1 Description......Page 294 28.3 Computing Table Values......Page 295 28.5 Relation to Other Distributions......Page 296 28.6 Random Number Generation......Page 297 29.1 Description......Page 298 29.2 Computing Table Values......Page 299 29.3 An Example......Page 300 29.4 Inferences on Correlation Coe±cients......Page 301 An Exact Test......Page 302 A Generalized Variable Test......Page 303 An Exact Confidence Interval......Page 304 The Generalized Confidence Interval for p......Page 305 An Asymptotic Approach......Page 306 Generalized Test and Confidence Limits for p1 --p2......Page 307 29.6 Random Number Generation......Page 308 29.7 A Computational Algorithm for Probabilities......Page 310 30.1 Description......Page 311 Moments......Page 312 30.3 Examples......Page 313 31.1 Hypothesis Test for the Median......Page 315 31.3 Computing Table Values......Page 316 31.4 An Example......Page 317 32.1 Description......Page 318 32.2 Moments and an Approximation......Page 319 32.4 An Example......Page 320 33.1 Description......Page 322 33.3 Mann-Whitney U Statistic......Page 323 33.5 An Example......Page 324 34.1 Description......Page 326 34.3 An Example......Page 327 35.1 Description......Page 328 35.3 Examples......Page 329 36.1 Description......Page 331 36.3.1 Point Estimation......Page 332 36.3.3 Hypothesis Testing......Page 333 36.6 A Computational Method for Probabilities......Page 334 36.7 Computing Table Values......Page 336 References......Page 337