Modern Probability Theory and Its Applications 0471668257
Mathematical probability theory is especially interesting to scientists and engineers. It introduces probability theory,
362 63 6MB
English Pages 480 Year 1960
Table of contents :
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
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
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