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Ross S. — A First Course in Probability
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Íàçâàíèå: A First Course in ProbabilityÀâòîð: Ross S. Àííîòàöèÿ: 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.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Èçäàíèå: Fifth editionÃîä èçäàíèÿ: 1998Êîëè÷åñòâî ñòðàíèö: 514Äîáàâëåíà â êàòàëîã: 30.03.2008Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
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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