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Ross S. — A First Course in Probability

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Название: A First Course in Probability

Автор: Ross S.

Аннотация:

This market leader is written as an elementary introduction to the mathematical theory of probability for readers in mathematics, engineering, and the sciences who possess the prerequisite knowledge of elementary calculus. A major thrust of the Fifth Edition has been to make the book more accessible to today's readers. The exercise sets have been revised to include more simple, "mechanical" problems and new section of Self-test Problems, with fully worked out solutions, conclude each chapter. In addition many new applications have been added to demonstrate the importance of probability in real situations. A software diskette, packaged with each copy of the book, provides an easy to use tool to derive probabilities for binomial, Poisson, and normal random variables. It also illustrates and explores the central limit theorem, works with the strong law of large numbers, and more.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Fifth edition

Год издания: 1998

Количество страниц: 514

Добавлена в каталог: 30.03.2008

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Предметный указатель

Antithetic variables      465—466
Associative laws for events      27
Axioms of probability      30—31
Axioms of surprise      437
Ballot problem      121
Banach match problem      165—166
Basic principle of counting      2
Basic principle of counting, generalized      3
Bayes’ formula      79
Bernoulli random variable      144
Bernoulli trials      120
Bernoulli, Daniel      315
Bernoulli, Jacob      see “James Bernoulli”
Bernoulli, James      93 398
Bernoulli, Nicholas      398
Bertrand’s paradox      203—204
Beta distribution      226 240 285
Binary symmetric channel      445
Binomial coefficients      8
Binomial random variable      144—145 150 172 185; 335—336 426
Binomial random variable, approximation to hypergeometric      168—169
Binomial random variable, computing its distribution function      152
Binomial random variable, randomly chosen success probability      346—347 363
Binomial random variable, simulation of      463—464
Binomial random variable, sums of independent      271 360—361
Binomial theorem      8
Birthday problem      40—41 157—158
Bivariate normal distribution      303—304 354 389—390
Bonferroni’s inequality      63 393
Boole’s inequality      66 312
Borel      411
Box — Muller      460
Branching process      390—391
Bridge      40
Buffon’s needle probem      255—236 300
Cauchy distribution      225 305
Cauchy distribution, standard      243
Cauchy — Schwarz inequality      387—388
Central limit theorem      204—205 399
Central limit theorem, for independent random variables      406
Channel capacity      449—450
Chapman — Kolmogorov equations      433
Chebyshev’s inequality      396
Chebyshev’s inequality, one-sided version      412—414
Chernoff bounds      415—417
Chi-squared distribution      267—268 361—362 367—368
Chi-squared distribution, relation to gamma distribution      267 284 301
Chi-squared distribution, simulation of      462
Coding theory      441—446
Combinations      6
Combinatorial analysis      2
Commulative laws for events      27
Complementary events      27 53
Complete graph      95
Conditional covariance formula      387
Conditional distribution      272 273
Conditional expectation      335 336 337
Conditional expectation, use in prediction      350—354
Conditional expectation, use in simulation      466
Conditional probability      67—68
Conditional probability distribution function      272 274
Conditional probability mass function      272 292
Conditional probability, as a long run relative frequency      72
Conditional probability, as a probability function      96—98
Conditional probability, density function      273—274 275 292
Conditional variance      348
Conditional variance formula      348
Conditionally independent events      102
Continuity property of probability      48—49 87
Continuous random variables      192
Control variates      468
Convex function      417
Convolution      265
Correlation      332—333
Correlation coefficient      see “Correlation”
Coupon collecting problems      129—131 315—316 386
Covariance      326—327 388
Craps      58
Cumulative distribution function      131 171
Cumulative distribution function, properties of      132—133
DeMere      89
DeMoivre      204 212 214—215 401
DeMoivre — Laplace limit theorem      212
DeMorgan’s laws      28—29
Dependent events      84
Dependent random variables      253
Deviations      328
Discrete random variables      134 171
Discrete uniform random variable      241—242
Distribution function      see “Cumulative distribution function”
Distributive laws for events      27
Dominant gene      112
Double exponential distribution      see “Laplacian distribution”
Ehrenfest urn model      432 436
entropy      439
Entropy, relation to coding theory      441
Ergodic Markov chain      434—435
Erlang distribution      224
Event      26
Exchangeable random variables      288—291 292
Expectation      136—137 171 185 309 368—370 384—385 388
Expectation, as a center of gravity      138—139
Expectation, of a beta random variable      227
Expectation, of a binomial random variable      149—150 313
Expectation, of a continuous random variable      195—196
Expectation, of a function of a random variable      139—140 310
Expectation, of a gamma random variable      224
Expectation, of a geometric random variable      163
Expectation, of a hypergeometric random variable      169—170 313—314
Expectation, of a negative binomial random variable      166 313
Expectation, of a nonnegative random variable      197
Expectation, of a normal random variable      206
Expectation, of a Poisson random variable      156
Expectation, of a sum of a random number of random variable      339—340
Expectation, of a uniform random variable      201
Expectation, of an exponential random variable      216
Expectation, of number of matches      314—315
Expectation, of sums of random variables      310—325
Expectation, tables of      359 360
Expected value      see “Expectation”
Exponential random variable      215 239 430
Exponential random variable, relation to, half-life, simulation of      456
Exponential random variable, sums of      267 286—288
Failure rate      see “Hazard rate”
Fermat      89—90 93
Fermat’s combinatorial identity      21
First moment      see “Mean”
Frequency interpretation of probability      30
Galton      407
Gambler’s ruin problem      90—93 120
Game theory      177
Gamma function      222—223 227 239
Gamma random variable      222 239 266—267 284—285 302—303
Gamma random variable, relation to chi-squared distribution      224 267 301
Gamma random variable, relation to exponential random variables      267
Gamma random variable, relation to Poisson process      223—224
Gamma random variable, simulation of      456
Gauss      214 215
Genetics      112 114 117
Geometric random variable      162 163 187 191
Geometric random variable, simulation of      463
Geometrical probability      203
Half-life      261—263
Hamiltonian permuation      323—324
Hazard rate      220—221 301
Huygens      89 93
Hypergeometric random variable      167 172 336
Hypergeometric random variable, relation to binomial      168—169 170
Importance sampling      471—472
Independent events      83—86

Independent increments      428
Independent random variables      252—253 257—258 259 263—264 292 301
information      439
Inheritance, theory of      147
Intersection of events      26 27 53
Inverse transform method of simulation      455
Jensen’s Inequality      418
Joint cumulative probability distribution function      244 251 291
Joint moment generating function      364
Joint probability density function      247—248 251—252 291
Jointly continuous random variables      247 251 291
k-of-n system      113
Keno      183—184
Khintchine      398
Kolmogorov      411
Laplace      204 212 401 407
Laplace distribution      219
Laplace’s rule of succession      102—103 122 123
Law of Large Numbers      395
Legendre theorem      240
Liapounoff      401
Limit of events      48
Linear prediction      353—354 389 390
Lognormal distribution      240 392
Marginal distribution      245 246
Markov chains      431—436
Markov’s inequality      395
Matching problem      44—45 63 100—101
Maximum likelihood estimates      168
Mean of a random variable      142
Median of a random variable      238—239
Memoryless random variable      217 218
Mendel      147
Midrange      304
Mode of a random variable      239
Moment generating function      355 358—359 392
Moment generating function, of a binomial random variable      356
Moment generating function, of a normal random variable      357—358
Moment generating function, of a Poisson rjandom variable      356—357
Moment generating function, of a sum of a random number of random variables      362—263
Moment generating function, of an exponential random variable      357
Moment generating function, tables for      359 360
Multinomial coefficients      11—12
Multinomial distribution      252 334—335
Multinomial theorem      12
Multiplication rule of probability      71
Multivariate normal distribution      365—366 392
Mutually exclusive events      27 53
Negative binomial random variable      164—165 172
Negative binomial random variable, relation to binomial      187
Negative binomial random variable, relation to geometric      165 301
Noiseless coding theorem      443
Noisy coding theorem      446
Normal random variable      204 282—284 364
Normal random variable, approximation to binomial      212
Normal random variable, characterization of      256—257
Normal random variable, joint distribution of sample mean and sample variance      366—368
Normal random variable, moments of      391
Normal random variable, simulation of      458—462
Normal random variable, sums of independent      268—269 292 361
Normal random variable, table for      208
Null set      54
Odds ratio      77—78
Order statistics      276—277 278 290—291 292
Paradox problem      50—52
Parallel system      87
PARETO      171
Partition      62
Pascal      89—90
Pascal random variable      see “Negative binomial random variable”
Pearson, Karl      214 215
Permutation      3—5
Personal probability      52—53
poisson      154
Poisson process      159—161 428—431
Poisson random variable      154 172 186 187 253—255 273 302—303 330 365 393 404 425
Poisson random variable, as an approximation      154—155 156—158 418—420
Poisson random variable, computing its distribution function      161—162
Poisson random variable, simulation of      464
Poisson random variable, sums of independent      270—271 292 361
Poker      39
Poker dice      57
Polya’s urn model      289—290
Prize problem      344—346
Probabilistic method      95—96
Probability density function      192
Probability density function, relation to cumulative distribution function      195
Probability mass function      134 171
Probability mass function, relation to cumulative distribution function      135
Problem of the points      80—90 165
Quadratic prediction      389
Quicksort algorithm      320—322
Random number      393 453
Random permuation      453—454 469
Random subset      259—261 454—455
Random variable      126 171
Random walk      318—320 433—434
Range of a random sample      279—280
Rayleigh density function      221 283
Record values      386
Rejection method of simulation      456—458
Relative frequency definition of probability      30
Riemann hypothesis      174
Riemann zeta function      171
Round robin tournament      122
runs      46—48 64 99—100 317—318
Sample mean      311—312 328 333—334
Sample mean, joint distribution of sample mean and sample variance      366—368
Sample median      278—279 304
Sample space      25 53
Sample variance      328—329
Sampling from a finite population      330—332
Sampling with replacement      58
Sequential drug test      93
Shannon      446
Signal processing      352—353
Signal to noise ratio      424
Simulation      453
St. Petersburg paradox      178
Standard deviation      144 172 424
Standard normal random variable      207 238
Standard normal random variable, distribution function      207
Standard normal random variable, distribution function, inequalities for      211
Stationary increments      428
Stieltjes integral      368—369
Stirling’s approximation      151
Stochastically larger      385
Strong Law of Large Numbers      407—408
Strong law of large numbers, proof of      408—410
Subjective probability      see “Personal probability”
Sum of ranks      377—378
Surprise      436—438
Trials      86
Triangular distribution      266
Uncertainty      439 440
Uncorrelated random variables      333
Uniform random variable      200—201 240 343—344 373 381
Union of events      26 27 53
Unit normal random variable      see “Standard normal random variable”
Variance      142—143 171 200 238
Variance, as a moment of inertia      144
Variance, of a binomial random variable      149—150 329
Variance, of a gamma random variable      224
Variance, of a geometric random variable      164 342—343
Variance, of a hypergeometric random variable      169—170 332
Variance, of a negative binomial random variable      166
Variance, of a normal random variable      206
Variance, of a Poisson random variable      156
Variance, of a sum of a random number of random variables      349
Variance, of a uniform random variable      202

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