Modern Probability Theory and Its Applications

Parzen, Emanuel. Modern probability theory and its applications(Wiley,1960) ......Page 3
Copyright ......Page 5
Preface ......Page 8
Contents ......Page 11
1 Probability theory as the study of random phenomena 1 ......Page 16
2 Probability theory as the study of mathematical models of random phenomena 5 ......Page 20
3 The sample description space of a random phenomenon 8 ......Page 23
4 Events 11 ......Page 26
5 The definition of probability as a function of events on a sample description space 17 ......Page 32
6 Finite sample description spaces 23 ......Page 38
7 Finite sample description spaces with equally likely descriptions 25 ......Page 40
8 Notes on the literature of probability theory 28 ......Page 43
1 Samples and n-tuples 32 ......Page 47
2 Posing probability problems mathematically 42 ......Page 57
3 The number of “successes” in a sample 51 ......Page 66
4 Conditional probability 60 ......Page 75
5 Unordered and partitioned samples—occupancy problems 67 ......Page 82
6 The probability of occurrence of a given number of events 76 ......Page 91
1 Independent events and families of events 87 ......Page 102
2 Independent trials 94 ......Page 109
3 Independent Bernoulli trials 100 ......Page 115
4 Dependent trials 113 ......Page 128
5 Markov dependent Bernoulli trials 128 ......Page 143
6 Markov chains 136 ......Page 151
1 The notion of a numerical-valued random phenomenon 148 ......Page 163
2 Specifying the probability law of a numerical-valued random phenomenon 151 ......Page 166
Appendix: The evaluation of integrals and sums 160 ......Page 175
3 Distribution functions 166 ......Page 181
4 Probability laws 176 ......Page 191
5 The uniform probability law 184 ......Page 199
6 The normal distribution and density functions 188 ......Page 203
7 Numerical H-tuple valued random phenomena 193 ......Page 208
1 The notion of an average 199 ......Page 214
2 Expectation of a function with respect to a probability law 203 ......Page 218
3 Moment-generating functions 215 ......Page 230
4 Chebyshev’s inequality 225 ......Page 240
5 The law of large numbers for independent repeated Bernoulli trials 228 ......Page 243
6 More about expectation 232 ......Page 247
1 The importance of the normal probability law 237 ......Page 252
2 The approximation of the binomial probability law by the normal and Poisson probability laws 239 ......Page 254
3 The Poisson probability law 251 ......Page 266
4 The exponential and gamma probability laws 260 ......Page 275
5 Birth and death processes 264 ......Page 279
1 The notion of a random variable 268 ......Page 283
2 Describing a random variable 270 ......Page 285
3 An example, treated from the point of view of numerical n-tuple valued random phenomena 276 ......Page 291
4 The same example treated from the point of view of random variables 282 ......Page 297
5 Jointly distributed random variables 285 ......Page 300
6 Independent random variables 294 ......Page 309
7 Random samples, randomly chosen points (geometrical probability), and random division of an interval 298 ......Page 313
8 The probability law of a function of a random variable 308 ......Page 323
9 The probability law of a function of random variables 316 ......Page 331
10 The joint probability law of functions of random variables 329 ......Page 344
11 Conditional probability of an event given a random variable. Conditional distributions 334 ......Page 349
1 Expectation, mean, and variance of a random variable 343 ......Page 358
2 Expectations of jointly distributed random variables 354 ......Page 369
3 Uncorrelated and independent random variables 361 ......Page 376
4 Expectations of sums of random variables 366 ......Page 381
5 The law of large numbers and the central limit theorem . 371 ......Page 386
6 The measurement signal-to-noise ratio of a random variable 378 ......Page 393
7 Conditional expectation. Best linear prediction 384 ......Page 399
1 The problem of addition of independent random variables 391 ......Page 406
2 The characteristic function of a random variable 394 ......Page 409
3 The characteristic function of a random variable specifies its probability law 400 ......Page 415
4 Solution of the problem of the addition of independent random variables by the method of characteristic functions 405 ......Page 420
5 Proofs of the inversion formulas for characteristic functions 408 ......Page 423
1 Modes of convergence of a sequence of random variables 414 ......Page 429
2 The law of large numbers 417 ......Page 432
3 Convergence in distribution of a sequence of random variables 424 ......Page 439
4 The central limit theorem 430 ......Page 445
5 Proofs of theorems concerning convergence in distribution 434 ......Page 449
Tables 441 ......Page 456
Answers to Odd-Numbered Exercises 447 ......Page 462
Index 459 ......Page 474
cover......Page 1