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in ANOVA, 573, 577, 600, 614 and, 198 vs. f test, 576
F test and, 589, 593, 600, 604 Weibull distribution and, 203 two-factor, 587, 593, 600
in mixed effects model, 593, 604 Exponential random variable(s) type I error in, 574
in random effects model, 580, Box~Muller transformation Full quadratic model, 695
593-594 and, 271
in regression, 681 cdf of, 199 G
Expected value expected value of, 198 Galton—Watson branching process,
conditional, 255 independence of, 242 281
of a continuous random mean of, 198 Gamma distribution
variable, 171 in order statistics, 272, 275 chi-squared distribution and, 200
covariance and, 247 pdf of, 198 definition of, 195
of a discrete random variable, 112 transformation of, 220, 267, 270 density function for, 195
of a function, 115, 245-246 variance of, 198 Erlang distribution and, 201
heavy-tailed distribution and, Exponential regression model, 721 estimators of parameters, 351,
114-115, 120 Exponential smoothing, 48 355, 358
of jointly distributed random Extreme outliers, 3941 exponential distribution and,
variables, 245 Extreme value distribution, 217 198-200
Law of Large Numbers and, 303 Poisson distribution and, 783
of a linear combination, 306 F standard, 195
of mean squares (see Expected _Factorial notation, 69 Weibull distribution and, 203
mean squares) Factorization theorem, 363 Gamma function
moment generating function Factors, 552 incomplete, 196, 217
and, 122, 175 Failure rate function, 230 properties of, 195
moments and, 121 Family of probability distributions, Gamma random variables, 195
in order statistics, 272-273, 277 104, 213 Geometric distribution, 143, 225
of sample mean, 277, 296 F distribution Geometric random variables, 143
of sample standard deviation, chi-squared distribution and, 323. Goodness-of-fit test
340, 379 definition of, 323 for composite hypotheses,
of sample total, 296 expected value of, 325 732, 741
of sample variance, 339 for model utility test, 649, 687, 709 definition of, 723
Experiment noncentral, 574-575 for homogeneity, 745-747
binomial, 128, 240, 724 pdf of, 324 for independence, 747-749
definition of, 52 Finite population correction factor, 141 simple, 724-730
double-blind, 523 Fisher information, 371 Grand mean, 555, 584
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Index 839
H Intersection of events model utility test and, 721
Half-normal plot, 220 definition of, 53 in Neyman-Pearson theorem, 470
Histogram multiplication rule for probability significance level and, 470, 471
bimodal, 18 of, 77-79 sufficiency and, 380
class intervals in, 15-17 Invariance principle, 357 tests, 475
construction of, 12-20 Inverse matrix, 712 Limiting relative frequency, 58, 59
density, 17-18 Linear combination
multimodal, 19 J distribution of, 309
Pareto diagram, 24 Jacobian, 267 expected value of, 306
for pmf, 103 Jensen’s inequality, 231 independence in, 306
symmetric, 19 Joint cumulative distribution variance of, 307
unimodal, 18 function, 282 Linear probabilistic model, 617, 627
Hodges-Lehmann estimator, 379 Jointly distributed random variables Linear regression
Homogeneity, 745~747 bivariate normal distribution additive model for, 614, 682, 705
Hyperexponential distribution, 229 of, 258-260 ANOVA in, 649, 699, 768
Hypergeometric distribution, conditional distribution of, confidence intervals in, 643, 656
138-141 253-263 correlation coefficient in,
and binomial distribution, 141 correlation coefficients for, 249 662-671
Hypergeometric random variable, covariance between, 248 definition of, 617
138-141 expected value of function of, degrees of freedom in, 631,
Hypothesis 245-246 685, 708
alternative, 426 independence of, 238-239 least squares estimates in,
composite, 732-741, 744 linear combination of, 306-312 625-636, 679
definition of, 426 in order statistics, 274-276 likelihood ratio test in, 721
errors in testing of, 428-434 pdf of (see Joint probability mies in, 631, 639
notation for, 426 density functions) model utility test in, 648, 687, 708
null, 426 pmf of (sce Joint probability mass parameters in, 617, 624-636, 682
research, 427 functions) percentage of explained variation
simple, 469 transformation of, 265-270 in, 633-634
Hypothetical population, 6 variance of function of, 252, 307 prediction interval in, 654, 658, 689
Joint marginal density function, 245 residuals in, 629, 674, 685
I Joint probability mass function, summary statistics in, 627
Inclusive inequalities, 136 233-234 sums of squares in, 631-636, 686
Incomplete beta function, 207 Joint probability table, 233 ratio in, 648, 669, 690
Incomplete gamma function, Line graph, 102-103
196-197, 217 K Location parameter, 217, 367
Independence K-out-of-n system, 153 Logistic distribution, 279
chi-squared test for, 749 Kruskal-Wallis test, 784-785 Logistic regression model
conditional distribution and, K-tuple, 68-69 contingency tables for, 749-751
257-258 definition of, 620-622
correlation coefficient L fit of, 650-651
and, 250 lag 1 autocorrelation coefficient, 674 mes in, 650
covariance and, 250, 252 Laplace distribution, 478 in multiple regression analysis, 699
of events, 84-88 Laplace’s rule of succession, 782 Logit function, 621, 650
of jointly distributed random Largest extreme value distribution, 228 Lognormal distribution, 205-205, 233
variables, 238-239, 241 Law of Large Numbers, 303-304, Lognormal random variables, 205-206
in linear combinations, 306-307 322-323, 376
mutual, 87 Law of total probability, 79 M
pairwise, 90, 94 Least squares estimates, 626, 645, Mann-Whitney test, 766-770
in simple random sample, 287 679, 683-684 Marginal distribution, 234, 236, 253
Independent variable, 614 Level « test, 433 Marginal probability density
Indicator variables, 696 Level of a factor, 552, 583, 593 functions, 236
Inductive reasoning, 6 Levene test, 562-563 Marginal probability mass
Inferential statistics, 5-6 Leverages, 714715 functions, 234
Inflection point, 180 Likelihood function, 354, 470,475 Matrices in regression analysis,
Intensity function, 156 Likelihood ratio 705-715
Interaction, 597-602, 603-606, chi-squared statistic for, 477 Maximum likelihood estimator
693-698 definition of, 470 for Bernoulli parameter, 377
Intercept, 214, 617, 627 mle and, 475 for binomial parameter, 377
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