Modern probability theory and its applications / Emanuel Parzen.
Series A Wiley publication in mathematical statisticsEditor: New York : Tokyo : Wiley ; Toppan, c1960Edición: Wiley international edDescripción: xv, 464 p. : il. ; 22 cmOtra clasificación: 60-01CHAPTER PAGE 1 PROBABILITY THEORY AS THE STUDY OF MATHEMATICAL MODELS OF RANDOM PHENOMENA [1] 1 Probability theory as the study of random phenomena [1] 2 Probability theory as the study of mathematical models of random phenomena [5] 3 The sample description space of a random phenomenon [8] 4 Events [11] 5 The definition of probability as a function of events on a sample description space [17] 6 Finite sample description spaces [23] 7 Finite sample description spaces with equally likely descriptions [25] 8 Notes on the literature of probability theory [28] 2 BASIC PROBABILITY THEORY [32] 1 Samples and n-tuples [32] 2 Posing probability problems mathematically [42] 3 The number of “successes” in a sample [51] 4 Conditional, probability [60] 5 Unordered and partitioned samples—occupancy problems [67] 6 The probability of occurrence of a given number of events [76] 3 INDEPENDENCE AND DEPENDENCE [87] 1 Independent events and families of events [87] 2 Independent trials [94] 3 Independent Bernoulli trials [100] 4 Dependent trials [113] 5 Markov dependent Bernoulli trials [128] 6 Markov chains [136] 4 NUMERICAL-VALUED RANDOM PHENOMENA [148] 1 The notion of a numerical-valued random phenomenon [148] 2 Specifying the probability law of a numerical-valued random phenomenon [151] Appendix: The evaluation of integrals and sums [160] 3 Distribution functions [166] 4 Probability laws [176] 5 The uniform probability law [184] 6 The normal distribution and density functions [188] 7 Numerical n-tuple valued random phenomena [193] 5 MEAN AND VARIANCE OF A PROBABILITY LAW [199] 1 The notion of an average [199] 2 Expectation of a function with respect to a probability law [203] 3 Moment-generating functions [215] 4 Chebyshev’s inequality [225] 5 The law of large numbers for independent repeated Bernoulli trials [228] 6 More about expectation [232] 6 NORMAL, POISSON, AND RELATED PROBABILITY LAWS [237] 1 The importance of the normal probability law [237] 2 The approximation of the binomial probability law by the normal and Poisson probability laws [239] 3 The Poisson probability law [251] 4 The exponential and gamma probability laws [260] 5 Birth and death processes [264] 7 RANDOM VARIABLES [268] 1 The notion of a random variable [268] 2 Describing a random variable [270] 3 An example, treated from the point of view of numerical n-tuple valued random phenomena [276] 4 The same example treated from the point of view of random variables [282] 5 Jointly distributed random variables [285] 6 Independent random variables [294] 7 Random samples, randomly chosen points (geometrical probability), and random division of an interval [298] 8 The probability law of a function of a random variable [308] 9 The probability law of a function of random variables [316] 10 The joint probability law of functions of random variables [329] 11 Conditional probability of an event given a random variable. Conditional distributions [334] 8 EXPECTATION OF A RANDOM VARIABLE [343] 1 Expectation, mean, and variance of a random variable [343] 2 Expectations of jointly distributed random variables [354] 3 Uncorrelated and independent random variables [361] 4 Expectations of sums of random variables [366] 5 The law of large numbers and the central limit theorem [371] 6 The measurement signal-to-noise ratio of a random variable [378] 7 Conditional expectation. Best linear prediction [384] 9 SUMS OF INDEPENDENT RANDOM VARIABLES [391] 1 The problem of addition of independent random variables [391] 2 The characteristic function of a random variable [394] 3 The characteristic function of a random variable specifies its probability law [400] 4 Solution of the problem of the addition of independent random variables by the method of characteristic functions [405] 5 Proofs of the inversion formulas for characteristic functions [408] 10 SEQUENCES OF RANDOM VARIABLES [414] 1 Modes of convergence of a sequence of random variables [414] 2 The law of large numbers [417] 3 Convergence in distribution of a sequence of random variables [424] 4 The central limit theorem [430] 5 Proofs of theorems concerning convergence in distribution [434] Tables [441] Answers to Odd-Numbered Exercises [447] Index [459]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 60 P276 (Browse shelf) | Available | A-3656 |
"Notes on the literature of probability theory": p. 28-31.
CHAPTER --
PAGE --
1 PROBABILITY THEORY AS THE STUDY OF MATHEMATICAL MODELS OF RANDOM PHENOMENA [1] --
1 Probability theory as the study of random phenomena [1] --
2 Probability theory as the study of mathematical models of random phenomena [5] --
3 The sample description space of a random phenomenon [8] --
4 Events [11] --
5 The definition of probability as a function of events on a sample description space [17] --
6 Finite sample description spaces [23] --
7 Finite sample description spaces with equally likely descriptions [25] --
8 Notes on the literature of probability theory [28] --
2 BASIC PROBABILITY THEORY [32] --
1 Samples and n-tuples [32] --
2 Posing probability problems mathematically [42] --
3 The number of “successes” in a sample [51] --
4 Conditional, probability [60] --
5 Unordered and partitioned samples—occupancy problems [67] --
6 The probability of occurrence of a given number of events [76] --
3 INDEPENDENCE AND DEPENDENCE [87] --
1 Independent events and families of events [87] --
2 Independent trials [94] --
3 Independent Bernoulli trials [100] --
4 Dependent trials [113] --
5 Markov dependent Bernoulli trials [128] --
6 Markov chains [136] --
4 NUMERICAL-VALUED RANDOM PHENOMENA [148] --
1 The notion of a numerical-valued random phenomenon [148] --
2 Specifying the probability law of a numerical-valued random phenomenon [151] --
Appendix: The evaluation of integrals and sums [160] --
3 Distribution functions [166] --
4 Probability laws [176] --
5 The uniform probability law [184] --
6 The normal distribution and density functions [188] --
7 Numerical n-tuple valued random phenomena [193] --
5 MEAN AND VARIANCE OF A PROBABILITY LAW [199] --
1 The notion of an average [199] --
2 Expectation of a function with respect to a probability law [203] --
3 Moment-generating functions [215] --
4 Chebyshev’s inequality [225] --
5 The law of large numbers for independent repeated Bernoulli trials [228] --
6 More about expectation [232] --
6 NORMAL, POISSON, AND RELATED PROBABILITY LAWS [237] --
1 The importance of the normal probability law [237] --
2 The approximation of the binomial probability law by the normal and Poisson probability laws [239] --
3 The Poisson probability law [251] --
4 The exponential and gamma probability laws [260] --
5 Birth and death processes [264] --
7 RANDOM VARIABLES [268] --
1 The notion of a random variable [268] --
2 Describing a random variable [270] --
3 An example, treated from the point of view of numerical n-tuple valued random phenomena [276] --
4 The same example treated from the point of view of random variables [282] --
5 Jointly distributed random variables [285] --
6 Independent random variables [294] --
7 Random samples, randomly chosen points (geometrical probability), and random division of an interval [298] --
8 The probability law of a function of a random variable [308] --
9 The probability law of a function of random variables [316] --
10 The joint probability law of functions of random variables [329] --
11 Conditional probability of an event given a random variable. Conditional distributions [334] --
8 EXPECTATION OF A RANDOM VARIABLE [343] --
1 Expectation, mean, and variance of a random variable [343] --
2 Expectations of jointly distributed random variables [354] --
3 Uncorrelated and independent random variables [361] --
4 Expectations of sums of random variables [366] --
5 The law of large numbers and the central limit theorem [371] --
6 The measurement signal-to-noise ratio of a random variable [378] --
7 Conditional expectation. Best linear prediction [384] --
9 SUMS OF INDEPENDENT RANDOM VARIABLES [391] --
1 The problem of addition of independent random variables [391] --
2 The characteristic function of a random variable [394] --
3 The characteristic function of a random variable specifies its probability law [400] --
4 Solution of the problem of the addition of independent random variables by the method of characteristic functions [405] --
5 Proofs of the inversion formulas for characteristic functions [408] --
10 SEQUENCES OF RANDOM VARIABLES [414] --
1 Modes of convergence of a sequence of random variables [414] --
2 The law of large numbers [417] --
3 Convergence in distribution of a sequence of random variables [424] --
4 The central limit theorem [430] --
5 Proofs of theorems concerning convergence in distribution [434] --
Tables [441] --
Answers to Odd-Numbered Exercises [447] --
Index [459] --
MR, 22 #3021
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