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Ash R. — Basic probability theory
Ash R. — Basic probability theory

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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: en

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
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Предметный указатель
$F$ distribution      see Distribution
$t$ 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, $F$      278
Distribution, $t$      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, $n$-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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