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Bayes factor, 400 Bayesian: hypothesis tests, 399 interval estimation, 397 point estimation, 394 predictive distributions, 401 Bayes s theorem, 17 Beta distribution, 133 Binomial coefficients, 21 Binomial distribution, 118 moment generating function of, 121 normal approximation to, 126, 129 Poisson approximation to, 128 Bivariate normal distribution, 140 Buffon s needle problem, 64 Calculation of probabilities, 12 Cauchy distribution, 132, 133 characteristic function of, 132 moment generating function of, 132 relation to uniform distribution, 133 Central limit theorem, 129 for binomial random variables, 129 proof of, 130 Central tendency, measures of, 94 Change of variables, 51, 63 Characteristic function, 90, 97 of Cauchy distribution, 132 Chebyshev s inequality, 93 Chi-square distribution: moment generating function of, 134 relationship to F and t distributions, 139 relationship to normal distribution, 134 tests involving, 233, 242 Chi-square test of goodness of fit, 242, 246, 252 Coefficient of: contingency, 250 correlation, 91 determination, 301 linear correlation, 289 Combinations, 20 Combinatorial analysis, 17 probability using, 22 Conditional: density, 58 distribution, 58 expectation, 93 moments, 93 probability, 14 variance, 93 Confidence interval estimates for: differences of means, 203 differences of proportions, 202 means in large samples, 200 means in small samples, 202 mean when population variance is unknown, 174 proportions, 202, 207 standard deviation, 208 variances, 204 variance ratios, 205 Conjugate prior distributions, 393 Contingency, coefficient of, 250 Contingency tables, 246 Continuous distribution function, 46 Convolutions, 56 Correlation: coefficient, 91, 140 generalized, 292 linear, 289 multiple, 293 probability interpretation of, 295 product-moment formula for, 290 rank, 293 sampling theory of, 298 table, 305 Counting, 17 Covariance, 91, 184 Cyclic pattern, in runs test, 361 Determination, coefficient of, 301 Discrete distribution function, 45 Discrete random variable, 44 Dispersion, measures of, 96 Distribution: Bayesian predictive, 401 beta, 133 binomial, 118 bivariate normal, 140 Cauchy, 132 chi-square, 134 conditional, 58 conjugate prior, 393 continuous, 46 of differences and sums, 169 discrete, 4 Fisher s F, 138 frequency, 175 gamma, 133 hypergeometric, 131 improper prior, 392 joint, 47 marginal, 48 multinomial, 131 normal, 122 of means, 163 of proportions, 166 of ratios of variances, 174 Poisson, 128 prior and posterior, 388 relationships among, F, chi-square, and t, 139 sampling, 163, 166, 169, 171, 174 Student s t, 136 uniform, 132 of variances, 171 Weibull, 141 Distribution functions: continuous, 46 discrete, 45 Marginal, 48, 49 Efficient estimates, 199 Estimates: confidence interval, 200, 202 efficient, 199 maximum likelihood, 206 unbiased, 199 Events, 10 independent, 14 mutually exclusive, 13 Expectation of random variables, 85 conditional, 93 F distribution, 138 relationship to chi-square and t distributions, 139 tests involving, 233 Fitting of data by theoretical distributions, 239 Fourier coefficients, 97 Fourier series, 97 Frequency: distributions, 175 histograms, 176 polygons, 176 Gamma distribution, 133 Generalized correlation coefficient, 292 Geometric probability, applications to, 60 Goodness of fit, 242, 246 Graeco-Latin squares, 335 Hypergeometric distribution, 131 Hypotheses tests (see Tests) Improper distributions, 392 Independent events, 14 Independent random variables, 47, 59 Joint density functions, 48 Joint distributions, 47 Kruskal-Wallis H test, 283 Kurtosis, 96, 97, 98 Latin squares, 334 Law of large numbers, 93 for Bernoulli trials, 122 Least squares: line, 275 parabola, 284 Linear correlation coefficient (see Correlation) Linear relationship, 292 Log-log graph paper, 283
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when sampling from a Poisson population, 391 Probability: calculation of, 12 calculating using combinatorial analysis, 22, 23 conditional, 14 distributions, 44 geometric, 60 subjective, 388 theorems on, 11 Probability distributions: continuous, 46 discrete, 44 Probability interpretation of correlation and regression, 295 Product-moment formula, 290 Proportions, sampling distribution of , 166 Quality control charts, 238 Random experiments, 10 Random variables: conditional expectation of, 93 continuous, 46 discrete, 44 expectation of, 85 independent, 47 Randomness runs test, 361 Rank correlation, 293, 365 Regression: least-squares line of, 275 multiple, 285 probability interpretation of, 295 sampling theory of, 297 Runs test for randomness, 361 applications of, 364 Sample mean: coding formula for, 178 computation of, 177 Sample moments, computation of, 177 Sample spaces, 10 Sample variance: coding formula for, 179 computation of, 177 Sampling distribution of: difference of means, 169 mean when population variance is unknown, 174 means, 163, 181 proportions, 166 ratios of variances, 174 sum of variances, 169 variances, 171 Sampling theory of: correlation, 298 regression, 297 Scatter diagram, 280 Sign test, 352 Skewness, 96, 97, 181 Standard deviation, 87 Standard error of estimate, 287 Student s t distribution, 136 relationship to F and chi-square distributions, 139 tests involving, 236 Subjective probability, 388 Tests: of difference of means, 227 of differences of proportions, 227 involving the chi-square distribution, 233, 242 involving the F distribution, 233 involving the Student s t distribution, 230 of means using normal distributions, 222 of proportions using normal distributions, 222 Theoretical distributions, fitting of data by, 239 Three-factor experiments, 329 Three-way classification, 329 Tree diagrams, 17 Trend pattern, in runs test, 362 Trend values, 301 Two-factor experiments, 330 with replication, 331 Two-way classification, 330 Unbiased estimates, 199 Uniform distribution, 132 Variables, change of, 51, 63 Variables, random (see Random variables) Variance, 87 computation of, for samples, 177 conditional, 93 sampling distribution of, 171 Variation: explained, 289 total, 289 unexplained, 292 Weibull distribution, 141
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Mann-Whitney U test, 354 sampling distribution of, 358 Marginal density function, 58 Marginal distribution function, 48, 61 Maximum likelihood estimates, 206 for mean of normal distribution, 206 for variance of normal distribution, 207 Means: computation of, for samples, 177 sampling distribution of, 163 Measures of central tendency, 94 Measures of dispersion, 96 Median, 94 Mode, 94 Modifications for unequal number of observations, 328 Moment generating function, 88, 96 of binomial distribution, 121 of Cauchy distribution, 132 of Poisson distribution, 129 of sums of independent random variables, 89 Moments, 88 computation of, 93 conditional, 93 for samples, 177 Multinomial distribution, 131 Multiple correlation coefficient, 293 Mutually exclusive events, 13 Nonlinear equations reducible to linear equations, 282 Nonlinear relationship, 292 Normal approximation to binomial distribution, 126, 129 Normal distribution, 122 bivariate, 140 One-factor experiments, 324 One-way classification, 324 Operating characteristic curve, 234, 251 Operating characteristic function, 235 Percentiles, 96 Permutations, 18 Poisson distribution, 128 moment generating function of, 129 Power function, 235 Prior and posterior distributions, 388 when sampling from a binomial population, 390 when sampling from a normal population, 391
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