An introduction to probability theory and its applications.
Series Wiley mathematical statistics seriesEditor: New York, Wiley [1950-68]Edición: 3rd edDescripción: 2 v. diagrs. 24 cmTema(s): ProbabilitiesOtra clasificación: *CODIGO*Vol. 1 Contents chapter Introduction: The Nature of Probability Theory . [1] 1. The Background [1] 2. Procedure [3] 3. “Statistical” Probability [4] 4. Summary [5] 5. Historical Note [6] I The Sample Space 1. The Empirical Background [7] 2. Examples [9] 3. The Sample Space. Events [13] 4. Relations among Events [14] 5. Discrete Sample Spaces [17] 6. Probabilities in Discrete Sample Spaces: Preparations [19] 7. The Basic Definitions and Rules [22] 8. Problems for Solution [24] II Elements of Combinatorial Analysis [26] 1. Preliminaries [26] 2. Ordered Samples [28] 3. Examples [31] 4. Subpopulations and Partitions [34] *5. Application to Occupancy Problems [38] *5a. Bose-Einstein and Fermi-Dirac Statistics [40] *5b. Application to Runs .. [42] 6. The Hypergeometric Distribution [43] 7. Examples for Waiting Times [47] 8. Binomial Coefficients [50] 9. Stirling’s Formula [52] Problems for Solution: [54] 10. Exercises and Examples [54] 11. Problems and Complements of a Theoretical Character [58] 12. Problems and Identities Involving Binomial Coefficients [63] *111 Fluctuations in Coin Tossing and Random Walks [67] 1. General Orientation. The Reflection Principle [68] 2. Random Walks: Basic Notions and Notations [73] 3. The Main Lemma [75] 4. Last Visits and Long Leads [78] *5. Changes of Sign [84] 6. An Experimental Illustration [86] 7. Maxima and First Passages [88] 8. Duality. Position of Maxima [91] 9. An Equidistribution Theorem [94] 10. Problems for Solution [95] ♦IV Combination of Events [98] 1. Union of Events [98] 2. Application to the Classical Occupancy Problem [101] 3. The Realization of m among N events [106] 4. Application to Matching and Guessing [107] 5. Miscellany [109] 6. Problems for Solution Ill V Conditional Probability. Stochastic Independence [114] 1. Conditional Probability [114] 2. Probabilities Defined by Conditional Probabilities. Urn Models [118] 3. Stochastic Independence [125] 4. Product Spaces. Independent Trials [128] ♦5. Applications to Genetics [132] *6. Sex-Linked Characters [136] ♦7. Selection [139] 8. Problems for Solution [140] VI The Binomial and the Poisson Distributions [146] 1. Bernoulli Trials [146] 2. The Binomial Distribution [147] 3. The Central Term and the Tails [150] 4. The Law of Large Numbers [152] 5. The Poisson Approximation [153] 6. The Poisson Distribution [156] 7. Observations Fitting the Poisson Distribution [159] 8. Waiting Times. The Negative Binomial Distribution [164] 9. The Multinomial Distribution [167] 10. Problems for Solution [169] VII The Normal Approximation to the Binomial Distribution [174] 1. The Normal Distribution [174] 2. Orientation: Symmetric Distributions [179] 3. The DeMoivre-Laplace Limit Theorem [182] 4. Examples [187] 5. Relation to the Poisson Approximation [190] *6. Large Deviations [192] 7. Problems for Solution [193] ♦VIII Unlimited Sequences of Bernoulli Trials [196] 1. Infinite Sequences of Trials [196] 2. Systems of Gambling [198] 3. The Borel-Cantelli Lemmas [200] 4. The Strong Law of Large Numbers [202] 5. The Law of the Iterated Logarithm [204] 6. Interpretation in Number Theory Language [208] 7. Problems for Solution [210] IX Random Variables; Expectation [212] 1. Random Variables [212] 2. Expectations [220] 3. Examples and Applications [223] 4. The Variance [227] 5. Covariance; Variance of a Sum [229] 6. Chebyshev’s Inequality [233] *7. Kolmogorov’s Inequality [234] *8. The Correlation Coefficient [236] 9. Problems for Solution [237] X Laws of Large Numbers [243] 1. Identically Distributed Variables [243] *2. Proof of the Law of Large Numbers [246] 3. The Theory of “Fair” Games [243] ♦4. The Petersburg Game [251] 5. Variable Distributions [253] ♦6. Applications to Combinatorial Analysis [256] ♦7. The Strong Law of Large Numbers [258] 8. Problems for Solution [261] Xi Integral Valued Variables. Generating Functions [264] 1. Generalities [264] 2. Convolutions [266] 3. Equalizations and Waiting Times in Bernoulli Trials [270] 4. Partial Fraction Expansions [275] 5. Bivariate Generating Functions [279] *6. The Continuity Theorem [280] 7. Problems for Solution [283] ♦XII Compound Distributions. Branching Processes [286] 1. Sums of a Random Number of Variables [286] 2. The Compound Poisson Distribution [288] 2a. Processes with Independent Increments [292] 3. Examples for Branching Processes [293] 4. Extinction Probabilities in Branching Processes [295] 5. The Total Progeny in Branching Processes [298] 6. Problems for Solution [301] XIII Recurrent Events. Renewal Theory [303] 1. Informal Preparations and Examples [303] 2. Definitions [307] 3. The Basic Relations [311] 4. Examples [313] 5. Delayed Recurrent Events. A General Limit Theorem [316] 6. The Number of Occurrences of S [320] ♦7. Application to the Theory of Success Runs [322] ♦8. More General Patterns [326] 9. Lack of Memory of Geometric Waiting Times [328] 10. Renewal Theory [329] ♦11. Proof of the Basic Limit Theorem [335] 12. Problems for Solution [338] XIV Random Walk and Ruin Problems [342] 1. General Orientation [342] 2. The Classical Ruin Problem [344] 3. Expected Duration of the Game [348] *4. Generating Functions for the Duration of the Game and for the First-Passage Times [349] *5. Explicit Expressions [352] 6. Connection with Diffusion Processes [354] *7. Random Walks in the Plane and Space [359] 8. The Generalized One-Dimensional Random Walk (Sequential Sampling) [363] 9. Problems for Solution [367] XV Markov Chains [372] 1. Definition [372] 2. Illustrative Examples [375] 3. Higher Transition Probabilities [382] 4. Closures and Closed Sets [384] 5. Classification of States [387] 6. Irreducible Chains. Decompositions [390] 7. Invariant Distributions [392] 8. Transient Chains [399] 9. Periodic Chains [404] 10. Application to Card Shuffling [406] *11. Invariant Measures. Ratio Limit Theorems [407] *12. Reversed Chains. Boundaries [414] 13. The General Markov Process [419] 14. Problems for Solution [424] *XVI Algebraic Treatment of Finite Markov Chains [428] 1. General Theory [428] 2. Examples [432] 3.. Random Walk with Reflecting Barriers [436] 4. Transient States; Absorption Probabilities [438] 5. Application to Recurrence Times [443] XVII The Simplest Time-Dependent Stochastic Processes [444] 1. General Orientation. Markov Processes [444] 2. The Poisson Process [446] 3. The Pure Birth Process [448] *4. Divergent Birth Processes [451] 5. The Birth and Death Process [454] 6. Exponential Holding Times [458] 7. Waiting Line and Servicing Problems [460] 8. The Backward (Retrospective) Equations [468] 9. General Processes [470] 10. Problems for Solution [478] Answers to Problems [483] Index [499]
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Vol. 2 has series: Wiley series in probability and mathematical statistics.
Vol. 1
Contents --
chapter --
Introduction: The Nature of Probability Theory . [1] --
1. The Background [1] --
2. Procedure [3] --
3. “Statistical” Probability [4] --
4. Summary [5] --
5. Historical Note [6] --
I The Sample Space --
1. The Empirical Background [7] --
2. Examples [9] --
3. The Sample Space. Events [13] --
4. Relations among Events [14] --
5. Discrete Sample Spaces [17] --
6. Probabilities in Discrete Sample Spaces: Preparations [19] --
7. The Basic Definitions and Rules [22] --
8. Problems for Solution [24] --
II Elements of Combinatorial Analysis [26] --
1. Preliminaries [26] --
2. Ordered Samples [28] --
3. Examples [31] --
4. Subpopulations and Partitions [34] --
*5. Application to Occupancy Problems [38] --
*5a. Bose-Einstein and Fermi-Dirac Statistics [40] --
*5b. Application to Runs .. [42] --
6. The Hypergeometric Distribution [43] --
7. Examples for Waiting Times [47] --
8. Binomial Coefficients [50] --
9. Stirling’s Formula [52] --
Problems for Solution: [54] --
10. Exercises and Examples [54] --
11. Problems and Complements of a Theoretical Character [58] --
12. Problems and Identities Involving Binomial Coefficients [63] --
*111 Fluctuations in Coin Tossing and Random Walks [67] --
1. General Orientation. The Reflection Principle [68] --
2. Random Walks: Basic Notions and Notations [73] --
3. The Main Lemma [75] --
4. Last Visits and Long Leads [78] --
*5. Changes of Sign [84] --
6. An Experimental Illustration [86] --
7. Maxima and First Passages [88] --
8. Duality. Position of Maxima [91] --
9. An Equidistribution Theorem [94] --
10. Problems for Solution [95] --
♦IV Combination of Events [98] --
1. Union of Events [98] --
2. Application to the Classical Occupancy Problem [101] --
3. The Realization of m among N events [106] --
4. Application to Matching and Guessing [107] --
5. Miscellany [109] --
6. Problems for Solution Ill --
V Conditional Probability. Stochastic Independence [114] --
1. Conditional Probability [114] --
2. Probabilities Defined by Conditional Probabilities. Urn Models [118] --
3. Stochastic Independence [125] --
4. Product Spaces. Independent Trials [128] --
♦5. Applications to Genetics [132] --
*6. Sex-Linked Characters [136] --
♦7. Selection [139] --
8. Problems for Solution [140] --
VI The Binomial and the Poisson Distributions [146] --
1. Bernoulli Trials [146] --
2. The Binomial Distribution [147] --
3. The Central Term and the Tails [150] --
4. The Law of Large Numbers [152] --
5. The Poisson Approximation [153] --
6. The Poisson Distribution [156] --
7. Observations Fitting the Poisson Distribution [159] --
8. Waiting Times. The Negative Binomial Distribution [164] --
9. The Multinomial Distribution [167] --
10. Problems for Solution [169] --
VII The Normal Approximation to the Binomial Distribution [174] --
1. The Normal Distribution [174] --
2. Orientation: Symmetric Distributions [179] --
3. The DeMoivre-Laplace Limit Theorem [182] --
4. Examples [187] --
5. Relation to the Poisson Approximation [190] --
*6. Large Deviations [192] --
7. Problems for Solution [193] --
♦VIII Unlimited Sequences of Bernoulli Trials [196] --
1. Infinite Sequences of Trials [196] --
2. Systems of Gambling [198] --
3. The Borel-Cantelli Lemmas [200] --
4. The Strong Law of Large Numbers [202] --
5. The Law of the Iterated Logarithm [204] --
6. Interpretation in Number Theory Language [208] --
7. Problems for Solution [210] --
IX Random Variables; Expectation [212] --
1. Random Variables [212] --
2. Expectations [220] --
3. Examples and Applications [223] --
4. The Variance [227] --
5. Covariance; Variance of a Sum [229] --
6. Chebyshev’s Inequality [233] --
*7. Kolmogorov’s Inequality [234] --
*8. The Correlation Coefficient [236] --
9. Problems for Solution [237] --
X Laws of Large Numbers [243] --
1. Identically Distributed Variables [243] --
*2. Proof of the Law of Large Numbers [246] --
3. The Theory of “Fair” Games [243] --
♦4. The Petersburg Game [251] --
5. Variable Distributions [253] --
♦6. Applications to Combinatorial Analysis [256] --
♦7. The Strong Law of Large Numbers [258] --
8. Problems for Solution [261] --
Xi Integral Valued Variables. Generating Functions [264] --
1. Generalities [264] --
2. Convolutions [266] --
3. Equalizations and Waiting Times in Bernoulli Trials [270] --
4. Partial Fraction Expansions [275] --
5. Bivariate Generating Functions [279] --
*6. The Continuity Theorem [280] --
7. Problems for Solution [283] --
♦XII Compound Distributions. Branching Processes [286] --
1. Sums of a Random Number of Variables [286] --
2. The Compound Poisson Distribution [288] --
2a. Processes with Independent Increments [292] --
3. Examples for Branching Processes [293] --
4. Extinction Probabilities in Branching Processes [295] --
5. The Total Progeny in Branching Processes [298] --
6. Problems for Solution [301] --
XIII Recurrent Events. Renewal Theory [303] --
1. Informal Preparations and Examples [303] --
2. Definitions [307] --
3. The Basic Relations [311] --
4. Examples [313] --
5. Delayed Recurrent Events. A General Limit Theorem [316] --
6. The Number of Occurrences of S [320] --
♦7. Application to the Theory of Success Runs [322] --
♦8. More General Patterns [326] --
9. Lack of Memory of Geometric Waiting Times [328] --
10. Renewal Theory [329] --
♦11. Proof of the Basic Limit Theorem [335] --
12. Problems for Solution [338] --
XIV Random Walk and Ruin Problems [342] --
1. General Orientation [342] --
2. The Classical Ruin Problem [344] --
3. Expected Duration of the Game [348] --
*4. Generating Functions for the Duration of the Game and for the First-Passage Times [349] --
*5. Explicit Expressions [352] --
6. Connection with Diffusion Processes [354] --
*7. Random Walks in the Plane and Space [359] --
8. The Generalized One-Dimensional Random Walk (Sequential Sampling) [363] --
9. Problems for Solution [367] --
XV Markov Chains [372] --
1. Definition [372] --
2. Illustrative Examples [375] --
3. Higher Transition Probabilities [382] --
4. Closures and Closed Sets [384] --
5. Classification of States [387] --
6. Irreducible Chains. Decompositions [390] --
7. Invariant Distributions [392] --
8. Transient Chains [399] --
9. Periodic Chains [404] --
10. Application to Card Shuffling [406] --
*11. Invariant Measures. Ratio Limit Theorems [407] --
*12. Reversed Chains. Boundaries [414] --
13. The General Markov Process [419] --
14. Problems for Solution [424] --
*XVI Algebraic Treatment of Finite Markov Chains [428] --
1. General Theory [428] --
2. Examples [432] --
3.. Random Walk with Reflecting Barriers [436] --
4. Transient States; Absorption Probabilities [438] --
5. Application to Recurrence Times [443] --
XVII The Simplest Time-Dependent Stochastic Processes [444] --
1. General Orientation. Markov Processes [444] --
2. The Poisson Process [446] --
3. The Pure Birth Process [448] --
*4. Divergent Birth Processes [451] --
5. The Birth and Death Process [454] --
6. Exponential Holding Times [458] --
7. Waiting Line and Servicing Problems [460] --
8. The Backward (Retrospective) Equations [468] --
9. General Processes [470] --
10. Problems for Solution [478] --
Answers to Problems [483] --
Index [499] --
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