Mathematical theory of probability and statistics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
New York u.a.
Acad. Pr.
1967
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| Ausgabe: | 2. print. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 694 S. |
Internformat
MARC
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| 245 | 1 | 0 | |a Mathematical theory of probability and statistics |
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| 264 | 1 | |a New York u.a. |b Acad. Pr. |c 1967 | |
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Datensatz im Suchindex
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| adam_text | CONTENTS
Preface ... v
CHAPTER I FUNDAMENTALS
A. The Basic Assumptions (Sections 1 5) 1
Introduction — Sequences of Observations. The Label Space —
Frequency. Chance — Randomness — The Collective
B. The Operations (Sections 6 10) 15
First Operation: Place Selection — Second Operation: Mixing.
Probability as Measure — Third Operation: Partition — Fourth
Operation: Combining — Additional Remarks on Independence
APPENDIX ONE: THE CONSISTENCY OF THE NOTION
OF THE COLLECTIVE. WALD S RESULTS 39
APPENDIX TWO: MEASURE THEORETICAL APPROACH
VERSUS FREQUENCY APPROACH 43
CHAPTER II GENERAL LABEL SPACE
A. Distribution Function (Discrete Case).
Measure Theoretical Approach (Sections 1 3) 50
Introduction — Cumulative Distribution Function for the Discrete
Case — Non Countable Label Space. Measure Theoretical
Approach
B. Non Countable Label Space. Frequency Approach
(Sections 4 7) 58
The Field of Definition of Probability in a Frequency Theory —
Basic Extension — The Field Fx — Distribution Function.
Riemann Stieltjes Integral. Probability Density
APPENDIX THREE: TORNIER S FREQUENCY THEORY ... 98
ix
x CONTENTS
CHAPTER III BASIC PROPERTIES OF DISTRIBUTIONS
A. Mean Value, Variance, and Other Moments
(Sections 1 4) 112
Mean Value and Variance. Tchebycheff s Inequality — Expecta¬
tion Relative to a Distribution. Stieltjes Integral — Generalizations
of Tchebycheff s Inequality — Moments of a Distribution
B. Gaussian Distribution, Poisson Distribution
(Sections 5 and 6) 130
The Normal or Gaussian Distribution in One Dimension —
The Poisson Distribution
C. Distributions in Rn (Sections 7 and 8) 140
Distributions in More Than One Dimension — Mean Value and
Variance in Several Dimensions
CHAPTER IV EXAMPLES OF COMBINED OPERATIONS
A. Uniform Distributions (Sections 1 and 2) 155
Uniform Arithmetical Distribution — Uniform Density. Needle
Problem
B. Bernoulli Problem and Related Questions (Sections 3 6) . . . 166
The Problem of Repeated Trials — Bernoulli s Theorem — The
Approximation to the Binomial Distribution in the Case of Rare
Events. Poisson Distribution — The Negative Binomial Distribu¬
tion
C. Some Problems of Non independent Events
(Sections 7 9) 184
A Problem of Runs — Arbitrarily Linked Events. Basic Relations
— Examples of Arbitrarily Linked Events
D. Application to Mendelian Heredity Theory
(Sections 10 and 11) 202
Basic Facts and Definitions — Probability Theory of Linkage
E. Comments On Markov Chains (Sections 12 and 13) .... 210
Definitions. Classification — Applications of Markov Chains
CONTENTS xi
CHAPTER V SUMMATION OF CHANCE VARIABLES
CHARACTERISTIC FUNCTION
A. Summation of Chance Variables and Laws of Large
Numbers (Sections 1 4) 224
Summation of Chance Variables — The Laws of Large Numbers
— Laws of Large Numbers Continued. Khintchine s Theorem.
Markov s Theorem — Strong Laws of Large Numbers
B. Characteristic Function (Sections 5 8) 243
The Characteristic Function — Inversion — Solution of the
Summation Problem. Stability of the Normal Distribution and of
the Poisson Distribution — Continuity Theorem for Characteristic
Functions
CHAPTER VI ASYMPTOTIC DISTRIBUTION OF THE SUM
OF CHANCE VARIABLES
A. Asymptotic Results for Infinite Products.
Stirling s Formula. Laplace s Formula (Sections 1 and 2) .... 263
Product of an Infinite Number of Functions — Application of the
Product Formulas
B. Limit Distribution of the Sum of Independent
Discrete Random Variables (Sections 3 and 4) 276
Arithmetical Probabilities — Examples
C. Probability Density. Central Limit Theorem.
Lindeberg s and Liapounoff s Conditions (Sections 5 7) .... 289
The Summation Problem in the General Case—The Central Limit
Theorem. Necessary and Sufficient Conditions — Liapounoff s
Sufficient Condition
D. Probability of the Sum of Rare Events.
Compound Poisson Distribution (Sections 8 10) 305
Asymptotic Distribution of the Sum of n Discrete Random
Variables in the Case of Rare Events — Limit Probability of the
Sum of Rare Events as a Compound Poisson Distribution —
A Generalization of the Theorem of Section 8
APPENDIX FOUR : REMARKS ON ADDITIVE TIME
DEPENDENT STOCHASTIC PROCESSES 319
xii CONTENTS
CHAPTER VII PROBABILITY INFERENCE.
BAYES METHOD
A. Inference from a Finite Number of Observations
(Sections 1 and 2) 329
Bayes Problem and Solution — Discussion of po(x). Assumption
po(x) = constant
B. Law of Large Numbers (Section 3) 339
Bayes Theorem. Irrelevance of po{x) for large n
C. Asymptotic Distributions (Sections 4 6) 345
Limit Theorems for Bayes Problem — Application of the Two
Basic Limit Theorems to the Theory of Errors — Inference on a
Statistical Function of Unknown Probabilities
D. Rare Events (Section 7) 362
Inference on the Probability of Rare Events
CHAPTER VIM MORE ON DISTRIBUTIONS
A. Sample Distribution and Statistical Parameters
(Section 1 3) 368
Repartition — Some Statistical Parameters — Expectations and
Variances of Sample Mean and Sample Variance
B. Moments. Inequalities (Sections 4 and 5) 384
Determining a Distribution by Its First (2m — 1) Moments —
Some Inequalities
C. Various Distributions Related to Normal Distributions
(Sections 6 and 7) 402
The Chi Square Distribution. Some Applications — Student s
Distribution and F Distribution
D. Multivariate Normal Distribution (Sections 8 and 9) .... 416
Normal Distribution in Three Dimensions — Properties of the
Multivariate Normal Distribution
CONTENTS xiii
CHAPTER IX ANALYSIS OF STATISTICAL DATA
A. Lexis Theory (Sections 1 and 2) 431
Repeated Equal Alternatives — Non Equal Alternatives
B. Student Test and F Test (Section 3) 441
The Two Tests
C. The X2 Test (Sections 4 and 5) 447
Checking a Known Distribution — X2 Test if Certain Parameters
of the Theoretical Distribution Are Estimated from the Sample
D. The w2 Tests (Sections 6 and 7) 471
von Mises Definition — Smirnov s co2 Test
E. Deviation Tests (Section 8) 490
On the Kolmogorov Smirnov Tests
CHAPTER X PROBLEM OF INFERENCE
A. Testing Hypotheses (Sections 1 4) 494
The Basis of Statistical Inference — Testing Hypotheses. Intro¬
duction of Neyman Pearson Method — Neyman Pearson Method.
Composite Hypothesis. Discontinuous and Multivariate Cases —
On Sequential Sampling
B. Global Statements on Parameters (Section 5) 537
Confidence Limits
C. Estimation (Sections 6 and 7) 547
Maximum Likelihood Method — Further Remarks on Estimation
CHAPTER XI MULTIVARIATE STATISTICS. CORRELATION
A. Measures of Correlation in Two Dimensions
(Sections 1 3) 566
Correlation — Regression Lines — Other Measures of Correlation
xiv CONTENTS
B. Distribution of the Correlation Coefficient
(Sections 4 and 5) 584
Asymptotic Expectation and Variance of the Correlation Coefficient
— The Distribution of r in Normal Samples
C. Generalizations to k Variables (Sections 6 and 7) 599
Regression and Correlation in k Variables — Remarks on the
Distribution of Correlation Measures from a ^ Dimensional
Normal Population
D. First Comments on Statistical Functions (Section 8) .... 610
Asymptotic Expectation and Variance of Statistical Functions
CHAPTER XII INTRODUCTION TO THE THEORY OF
STATISTICAL FUNCTIONS
A. Differentiable Statistical Functions (Sections 1 and 2) .... 615
Statistical Functions. Continuity, Differentiability — Higher
Derivatives. Taylor s Theorem
B. The Laws of Large Numbers (Sections 3 and 4) 630
The First Law of Large Numbers for Statistical Functions — The
Second Law of Large Numbers for Statistical Functions
C. Statistical Functions of Type One (Sections 5 and 6) .... 641
Convergence Toward the Normal Distribution — Convergence
toward the Gaussian Distribution. General Case
D. Classification of Differentiable Statistical Functions
(Sections 7 and 8) 657
Asymptotic Expressions for Expectations — Asymptotic Behavior of Statistical
Functions
Selected Reference Books 669
Tables 670
Index 687
|
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| discipline | Mathematik |
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| format | Book |
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| physical | XVI, 694 S. |
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| spelling | Von Mises, Richard 1883-1953 Verfasser (DE-588)119161303 aut Mathematical theory of probability and statistics 2. print. New York u.a. Acad. Pr. 1967 XVI, 694 S. txt rdacontent n rdamedia nc rdacarrier Theorie (DE-588)4059787-8 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Statistik (DE-588)4056995-0 s Theorie (DE-588)4059787-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002363478&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Von Mises, Richard 1883-1953 Mathematical theory of probability and statistics Theorie (DE-588)4059787-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4059787-8 (DE-588)4064324-4 (DE-588)4056995-0 (DE-588)4151278-9 |
| title | Mathematical theory of probability and statistics |
| title_auth | Mathematical theory of probability and statistics |
| title_exact_search | Mathematical theory of probability and statistics |
| title_full | Mathematical theory of probability and statistics |
| title_fullStr | Mathematical theory of probability and statistics |
| title_full_unstemmed | Mathematical theory of probability and statistics |
| title_short | Mathematical theory of probability and statistics |
| title_sort | mathematical theory of probability and statistics |
| topic | Theorie (DE-588)4059787-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Statistik (DE-588)4056995-0 gnd |
| topic_facet | Theorie Wahrscheinlichkeitsrechnung Statistik Einführung |
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