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 Главная    Ex Libris    Книги    Журналы    Статьи    Серии    Каталог    Wanted    Загрузка    ХудЛит    Справка    Поиск по индексам    Поиск    Форум Авторизация Поиск по указателям     Ash R. — Basic probability theory Обсудите книгу на научном форуме Нашли опечатку?Выделите ее мышкой и нажмите Ctrl+Enter Название: Basic probability theory Автор: Ash R. Аннотация: Geared toward advanced undergraduates and graduate students, this introductory text surveys random variables, conditional probability and expectation, characteristic functions, infinite sequences of random variables, Markov chains, and an introduction to statistics. Complete solutions to some of the problems appear at the end of the book. 1970 edition. Язык: Рубрика: Математика/ Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2008 Количество страниц: 348 Добавлена в каталог: 17.02.2014 Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID Предметный указатель distribution      see Distribution distribution      see Distribution Absolutely continuous random variable      53 Absolutely continuous random vector      72 Actions, set of      242 Admissible and inadmissible tests      247 Admissible risk points      248 Alternative, simple and composite      243 Average value      see Expectation Bayes estimate      260 Bayes estimate, with constant risk      262 Bayes estimate, with quadratic loss function      260 Bayes risk      244 260 Bayes test      244 Bayes’ Theorem      36 150 Bernoulli distribution      see Distribution Bernoulli trials      28 38 58 128 151 175 177 187 190 195 207 215 Bernoulli trials, generalized      29 (see also Distribution binomial) Beta distribution      see Distribution beta function      133 261 Binomial distribution      see Distribution Boolean algebra      3ff Borel measurable function      83 Borel sets      47 50 Borel — Cantelli lemma      205 Borel — Cantelli lemma, second      209 Bose — Einstein assumption      21 Cauchy distribution      see Distribution Central limit theorem      169ff 171 Characteristic function(s)      154ff Characteristic function(s), correspondence, theorem for      156 Characteristic function(s), of a random vector      279 Characteristic function(s), properties of      166ff Chebyshev’s inequality      126 127 129 206 208 coin tossing      see Bernoulli trials; Distribution binomial Combinatorial problems      15ff Combinatorial problems, fallacies in      39ff Combinatorial problems, multiple counting in      22 Complement of an event      4 Conditional density      136 148 Conditional distribution function      139 140 148 Conditional expectation      140ff Conditional probability      33ff 130ff Conditional probability function      98 142 Confidence coefficient      276 Confidence interval      276 Confidence set      278 Continuous random variable      69 Convergence, almost surely (almost everywhere)      204—206 208 210 Convergence, in distribution      170 171 175 176 Convergence, in probability      171 175 176 205 208 210 Convex function      262 Convexity of the risk set      248 Convolution theorem      164 Correlation      119ff Correlation coefficient      120 Covariance      119 Covariance function      203 Covariance matrix      281 Cylinder      180 Cylinder, measurable      180 Decision function      242 243 Decision function, nonrandomized      242 Decision scheme      151 DeMorgan laws      7 9 11 Density function(s)      53 Density function(s), conditional      136 148 Density function(s), joint      70ff 181 Density function(s), marginal      78 Difference equation      24 39 182 186 195 Difference equation, characteristic equation of      183 Discrete probability space      15 Discrete random variables      51 95ff Disjoint events      5 distribution      95 Distribution function(s)      52 Distribution function(s), conditional      139 140 148 Distribution function(s), joint      72 Distribution function(s), properties of      66ff Distribution, 278 Distribution, 277 278 Distribution, Bernoulli      256 264 266 269 272 Distribution, Beta      260 268 Distribution, binomial      29 32 95 97—99 113 122 141 176 256 258 260 264 268 Distribution, Cauchy      161 166 264 Distribution, chi-square      165 275 276 278 Distribution, exponential      56 65 93 110 111 113 129 152 166 168 196 200— 202 256 263 264 Distribution, gamma      166 267 268 Distribution, geometric      195 196 Distribution, hypergeometric      33 256 Distribution, multidimensional Gaussian (joint Gaussian)      279ff Distribution, negative binomial      196 264 268 272 Distribution, normal (Gaussian)      87 88 92 94 108 113 118 124—126 162 165 166 171 173—176 252 256 267 268 271 274—276 278 Distribution, Poisson      96—99 114 152 163 169 197 198 200 202 256 264 266 268 270 272 Distribution, uniform      54 73 76 84 92 93 113 118 141 149 150— 152 165 208 257 263 264 267 271 272 Dominated Convergence Theorem      231 Essentially constant random variable      85 115 Estimate      258 Estimate, Bayes      260 Estimate, Bayes, with constant risk      262 Estimate, inadmissible      272 Estimate, maximum likelihood      258 Estimate, minimax      262 Estimate, randomized      258 263 Estimate, risk function of      261 Estimate, unbiased      268 Estimate, uniformly minimum variance unbiased (UMVUE)      269 Estimation      152 242 243 258ff Event(s)      2 11 Event(s), algebra of      3ff Event(s), complement of      4 Event(s), contracting sequence of      67 Event(s), exhaustive      35 Event(s), expanding sequence of      66 Event(s), impossible      3 55 Event(s), independent      26 27 Event(s), intersection of      4 Event(s), mutually exclusive (disjoint)      5 Event(s), sure (certain)      3 Event(s), union of      4 Event(s), upper and lower limits of sequence of      204 209 Expectation      100ff Expectation, conditional      140ff Expectation, general definition of      103 Expectation, properties of      114ff Exponential distribution      see Distribution Factorization theorem      266 Fatou’s Lemma      230 Fermi — Dirac assumption      20 Fourier series      167 Fourier transform      155 Gambler’s ruin problem      182ff 235 Gamma distribution      see Distribution Gamma function      109 133 Gaussian distribution      see Distribution normal Generating function      169 191ff Generating function, moments obtained from      192 Geometric distribution      see Distribution Hypergeometric distribution      see Distribution Hypothesis      243ff Hypothesis testing      151 242 243ff Hypothesis testing, fundamental theorem of      246 (see also Test) Hypothesis, a priori probability of      244 Hypothesis, composite      243 Hypothesis, null      243 Hypothesis, simple      243 Independence      25ff Independence of sample mean and variance in normal sampling      274 Independent events      26 27 Independent random variables      80 Indicators      122ff Intersection of events      4 Jensen’s Inequality      262 Joint characteristic function      279 Joint density function      70ff 181 Joint distribution function      72 Joint probability function      76 96 180 181 Kolmogorov Extension Theorem      180 Laplace transform      155 Laplace transform, properties of      156 157 Lattice distribution      169 Law of Large Numbers, strong      129 203 206 207 Law of large numbers, weak      128 169 171 207 Lebesgue integral      114 Level of a test      246 Liapounov condition      175 Likelihood ratio      245 Likelihood ratio, test (LRT)      245 Limit inferior (lower limit)      204 209 Limit superior (upper limit)      204 209 Linearly dependent random variables      121 281 Loss function (cost function)      242 Loss function (cost function), quadratic      260 Marginal densities      78 Markov chain(s)      211ff Markov chain(s), closed sets of      224 Markov chain(s), cyclically moving subclasses of      227 Markov chain(s), definition of      214 Markov chain(s), equivalence classes of states of      223 Markov chain(s), first entrance theorem for      220 Markov chain(s), initial distribution of      213 Markov chain(s), limiting probabilities of      2 30ff Markov chain(s), state distribution of      214 Markov chain(s), state space of      213 Markov chain(s), states of      220ff Markov chain(s), states of, aperiodic, periodic      229 Markov chain(s), states of, essential      229 Markov chain(s), states of, mean recurrence time of      230 Markov chain(s), states of, period of      226—229 Markov chain(s), states of, recurrent (persistent)      221ff Markov chain(s), states of, recurrent null      233 Markov chain(s), states of, recurrent positive      233 Markov chain(s), states of, transient      221ff Markov chain(s), stationary distribution for      236 Markov chain(s), steady state distribution for      215 237 Markov chain(s), stopping time for      217 Markov chain(s), strong Markov property of      219 Markov chain(s), transition matrix of      214 Markov chain(s), transition matrix of, -step      214 Markov chain(s), transition probabilities of      214 Maximum likelihood estimate      258 Maxwell — Boltzmann assumption      20 Mean      see Expectation Median      112 Minimax estimate      262 Minimax test      250 Moment-generating property of characteristic functions      167 168 Moments      107 Moments, central      108 Moments, joint      119 Moments, obtained from generating functions      192 Multinomial probability function      30 98 Mutually exclusive events      5 Negative binomial distribution      see Distribution Negative part of a random variable      104 Neyman — Pearson lemma      246 Normal distribution      see Distribution Observable      242 Order statistics      91 Partial fraction expansion      159 Poisson distribution      see Distribution Poisson random process      196ff Poker      19 23 40 Positive part of a random variable      104 Power function of a test      253 Power of a test      246 Probability function      51 Probability function, conditional      98 142 Probability function, joint      76 96 180 181 Probability measure(s)      12 Probability measure(s), consistent      180 Probability measure(s), discrete      15 Probability space      12 Probability, classical definition of      1 13 16 Probability, conditional      33ff Probability, frequency definition of      2 13 Probability, lOff Probability, a posteriori      36 Queueing      216 Random process      196 Random telegraph signal      203 Random variable(s)      46ff Random variable(s), absolutely continuous      53 Random variable(s), central moments of      108 Random variable(s), characteristic function of      154ff Random variable(s), classification of      51ff Random variable(s), continuous      69 Random variable(s), definition of      48 50 Random variable(s), degenerate (essentially constant)      85 115 Random variable(s), density function of      53 Random variable(s), discrete      51 95ff Random variable(s), functions of      58ff 84 85ff 94 Random variable(s), generating function of      192ff Random variable(s), independent      80 Random variable(s), infinite sequences of      178ff Random variable(s), linearly dependent      121 281 Random variable(s), moments of      107 Random variable(s), positive and negative parts of      104 Random variable(s), probability function of      51 Random variable(s), simple      101 Random vector      72 Random vector, absolutely continuous      72 Random walk      184ff Random walk, combinatorial approach to      186ff Random walk, simple      184 Random walk, simple, with absorbing barriers      184 185 215 228 240 Random walk, simple, with no barriers      185 186—191 193 195 215 228 240 Random walk, simple, with no barriers, average length of time required to return to 0 in      186 191 195 Random walk, simple, with no barriers, distribution of first return to 0 in      189 Random walk, simple, with no barriers, first passage times in      190 Random walk, simple, with no barriers, probability of eventual return to 0 in      185 Random walk, simple, with reflecting barriers      229 240 Rao — Blackwell theorem      263 Recurrent (persistent) states of a Markov chain      221 Reflection principle      188 Renewal Theorem      235 Risk function      261 Risk set      248 Sample mean      259 274 Sample space      2 Sample variance      259 274 Samples      16ff Samples, ordered, with replacement      16 Samples, ordered, without replacement      16 Samples, unordered, with replacement      18 Samples, unordered, without replacement      17 Sampling from a normal population      274 Schwarz inequality      119 121 207 Sigma field      11 Simple random variable      101 Size of a test      246 Standard deviation      108 States of nature      241 Statistic, for a random variable      265 Statistic, for a random variable, complete      269 Statistic, for a random variable, sufficient      265 Statistical decision model      241 Statistics      241ff Stirling’s formula      43 191 Stochastic matrix      212 Stochastic process      196
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