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**Author:**William Feller**Title:**An Introduction to Probability Theory and Its Applications**Amazon Link:**

https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

https://www.amazon.com/dp/0471257095/?tag=pfamazon01-20**Prerequisities:**

**Table of Contents for Volume I:**

Code:

```
[LIST]
[*] Introduction: The Nature of Probability Theory
[LIST]
[*] The Background
[*] Procedure
[*] "Statistical" Probability
[*] Summary
[*] Historical Note
[/LIST]
[*] The Sample Space
[LIST]
[*] The Empirical Background
[*] Examples
[*] The Sample Space. Events
[*] Relations among Events
[*] Discrete Sample Spaces
[*] Probabilities in Discrete Sample Spaces: Preparations
[*] The Basic Definitions and Rules
[*] Problems for Solution
[/LIST]
[*] Elements of Combinatorial Analysis
[LIST]
[*] Preliminaries
[*] Ordered Samples
[*] Examples
[*] Subpopulations and Partitions
[*] Application to Occupancy Problems
[LIST]
[*] Bose-Einstein and Fermi-Dirac Statistics
[*] Application to Runs
[/LIST]
[*] The Hypergeometric Distribution
[*] Examples for Waiting Times
[*] Binomial Coefficients
[*] Stirling's Formula
[*] Problems for Solution:
[LIST]
[*] Exercises and Examples
[*] Problems and Complements of a Theoretical Character
[*] Problems and Identities Involving Binomial Coefficients
[/LIST]
[/LIST]
[*] Fluctuations in Coin Tossing and Random Walks
[LIST]
[*] General Orientation. The Reflection Principle
[*] Random Walks: Basic Notions and Notations
[*] The Main Lemma
[*] Last Visits and Long Leads
[*] Changes of Sign
[*] An Experimental Illustration
[*] Maxima and First Passages
[*] Duality. Position of Maxima
[*] An Equidistribution Theorem
[*] Problems for Solution
[/LIST]
[*] Combination of Events
[LIST]
[*] Union of Events
[*] Application to the Classical Occupancy Problem
[*] The Realization of m among N events
[*] Application to Matching and Guessing
[*] Miscellany
[*] Problems for Solution
[/LIST]
[*] Conditional Probability. Stochastic Independence
[LIST]
[*] Conditional Probability
[*] Probabilities Defined by Conditional Probabilities. Urn Models
[*] Stochastic Independence
[*] Product Spaces. Independent Trials
[*] Applications to Genetics
[*] Sex-Linked Characters
[*] Selection
[*] Problems for Solution
[/LIST]
[*] The Binomial and the Poisson Distributions
[LIST]
[*] Bernoulli Trials
[*] The Binomial Distribution
[*] The Central Term and the Tails
[*] The Law of Large Numbers
[*] The Poisson Approximation
[*] The Poisson Distribution
[*] Observations Fitting the Poisson Distribution
[*] Waiting Times. The Negative Binomial Distribution
[*] The Multinomial Distribution
[*] Problems for Solution
[/LIST]
[*] The Normal Approximation to the Binomial Distribution
[LIST]
[*] The Normal Distribution
[*] Orientation: Symmetric Distributions
[*] The DeMoivre-Laplace Limit Theorem
[*] Examples
[*] Relation to the Poisson Approximation
[*] Large Deviations
[*] Problems for Solution
[/LIST]
[*] Unlimited Sequences of Bernoulli Trials
[LIST]
[*] Infinite Sequences of Trials
[*] Systems of Gambling
[*] The Borel-Cantelli Lemmas
[*] The Strong Law of Large Numbers
[*] The Law of the Iterated Logarithm
[*] Interpretation in Number Theory Language
[*] Problems for Solution
[/LIST]
[*] Random Variables; Expectation
[LIST]
[*] Random Variables
[*] Expectations
[*] Examples and Applications
[*] The Variance
[*] Covariance; Variance of a Sum
[*] Chebyshev's Inequality
[*] Kolmogorov's Inequality
[*] The Correlation Coefficient
[*] Problems for Solution
[/LIST]
[*] Laws of Large Numbers
[LIST]
[*] Identically Distributed Variables
[*] Proof of the Law of Large Numbers
[*] The Theory of "Fair" Games
[*] The Petersburg Game
[*] Variable Distributions
[*] Applications to Combinatorial Analysis
[*] The Strong Law of Large Numbers
[*] Problems for Solution
[/LIST]
[*] Integral Valued Variables. Generating Functions
[LIST]
[*] Generalities
[*] Convolutions
[*] Equalizations and Waiting Times in Bernoulli Trials
[*] Partial Fraction Expansions
[*] Bivariate Generating Functions
[*] The Continuity Theorem
[*] Problems for Solution
[/LIST]
[*] Compound Distributions. Branching Processes
[LIST]
[*] Sums of a Random Number of Variables
[*] The Compound Poisson Distribution
[LIST]
[*] Processes with Independent Increments
[/LIST]
[*] Examples for Branching Processes
[*] Extinction Probabilities in Branching Processes
[*] The Total Progeny in Branching Processes
[*] Problems for Solution
[/LIST]
[*] Recurrent Events. Renewal Theory
[LIST]
[*] Informal Preparations and Examples
[*] Definitions
[*] The Basic Relations
[*] Examples
[*] Delayed Recurrent Events. A General Limit Theorem
[*] The Number of Occurrences of [itex]\mathcal{E}[/itex]
[*] Application to the Theory of Success Runs
[*] More General Patterns
[*] Lack of Memory of Geometric Waiting Times
[*] Renewal Theory
[*] Proof of the Basic Limit Theorem
[*] Problems for Solution
[/LIST]
[*] Random Walk and Ruin Problems
[LIST]
[*] General Orientation
[*] The Classical Ruin Problem
[*] Expected Duration of the Game
[*] Generating Functions for the Duration of the Game and for the First-Passage Times
[*] Explicit Expressions
[*] Connection with Diffusion Processes
[*] Random Walks in the Plane and Space
[*] The Generalized One-Dimensional Random Walk (Sequential Sampling)
[*] Problems for Solution
[/LIST]
[*] Markov Chains
[LIST]
[*] Definition
[*] Illustrative Examples
[*] Higher Transition Probabilities
[*] Closures and Closed Sets
[*] Classification of States
[*] Irreducible Chains. Decompositions
[*] Invariant Distributions
[*] Transient Chains
[*] Periodic Chains
[*] Application to Card Shuffling
[*] Invariant Measures. Ratio Limit Theorems
[*] Reversed Chains. Boundaries
[*] The General Markov Process
[*] Problems for Solution
[/LIST]
[*] Algebraic Treatment of Finite Markov Chains
[LIST]
[*] General Theory
[*] Examples
[*] Random Walk with Reflecting Barriers
[*] Transient States; Absorption Probabilities
[*] Application to Recurrence Times
[/LIST]
[*] The Simplest Time-Dependent Stochastic Processes
[LIST]
[*] General Orientation. Markov Processes
[*] The Poisson Process
[*] The Pure Birth Process
[*] Divergent Birth Processes
[*] The Birth and Death Process
[*] Exponential Holding Times
[*] Waiting Line and Servicing Problems
[*] The Backward (Retrospective) Equations
[*] General Processes
[*] Problems for Solution
[/LIST]
[*] Answers to Problems
[*] Index
[/LIST]
```

**Table of Contents for Volume II:**

Code:

```
[LIST]
[*] The Exponential and the Uniform Densities
[LIST]
[*] Introduction
[*] Densities. Convolutions
[*] The Exponential Density
[*] Waiting Time Paradoxes. The Poisson Process
[*] The Persistence of Bad Luck
[*] Waiting Times and Order Statistics
[*] The Uniform Distribution
[*] Random Splittings
[*] Convolutions and Covering Theorems
[*] Random Directions
[*] The Use of Lebesgue Measure
[*] Empirical Distributions
[*] Problems for Solution
[/LIST]
[*] Special Densities. Randomization
[LIST]
[*] Notations and Conventions
[*] Gamma Distributions
[*] Related Distributions of Statistics
[*] Some Common Densities
[*] Randomization and Mixtures
[*] Discrete Distributions
[*] Bessel Functions and Random Walks
[*] Distributions on a Circle
[*] Problems for Solution
[/LIST]
[*] Densities in Higher Dimensions. Normal Densities and Processes
[LIST]
[*] Densities
[*] Conditional Distributions
[*] Return to the Exponential and the Uniform Distributions
[*] A Characterization of the Normal Distribution
[*] Matrix Notation. The Covariance Matrix
[*] Normal Densities and Distributions
[*] Stationary Normal Processes
[*] Markovian Normal Densities
[*] Problems for Solution
[/LIST]
[*] Probability Measures and Spaces
[LIST]
[*] Baire Functions
[*] Interval Functions and Integrals in [itex]\mathcal{R}^r[/itex]
[*] [itex]\sigma[/itex]-Algebras. Measurability
[*] Probability Spaces. Random Variables
[*] The Extension Theorem
[*] Product Spaces. Sequences of Independent Variables
[*] Null Sets. Completion
[/LIST]
[*] Probability Distributions in [itex]\mathcal{R}^r[/itex]
[LIST]
[*] Distributions and Expectations
[*] Preliminaries
[*] Densities
[*] Convolutions
[*] Symmetrization
[*] Integration by Parts. Existence of Moments
[*] Chebyshev's Inequality
[*] Further Inequalities. Convex Functions
[*] Simple Conditional Distributions. Mixtures
[*] Conditional Distributions
[*] Conditional Expectations
[*] Problems for Solution
[/LIST]
[*] A Survey of some Important Distributions and Processes
[LIST]
[*] Stable Distributions in [itex]\mathcal{R}^1[/itex]
[*] Examples
[*] Infinitely Divisible Distributions in [itex]\mathcal{R}^1[/itex]
[*] Processes with Independent Increments
[*] Ruin Problems in Compound Poisson Processes
[*] Renewal Processes
[*] Examples and Problems
[*] Random Walks
[*] The Queuing Process
[*] Persistent and Transient Random Walks
[*] General Markov Chains
[*] Martingales
[*] Problems for Solution
[/LIST]
[*] Laws of Large Numbers. Applications in Analysis
[LIST]
[*] Main Lemma and Notations
[*] Bernstein Polynomials. Absolutely Monotone Functions
[*] Moment Problems
[*] Application to Exchangeable Variables
[*] Generalized Taylor Formula and Semi-Groups
[*] Inversion Formulas for Laplace Transforms
[*] Laws of Large Numbers for Identically Distributed Variables
[*] Strong Laws
[*] Generalization to Martingales
[*] Problems for Solution
[/LIST]
[*] The Basic Limit Theorems
[LIST]
[*] Convergence of Measures
[*] Special Properties
[*] Distributions as Operators
[*] The Central Limit Theorem
[*] Infinite Convolutions
[*] Selection Theorems
[*] Ergodic Theorems for Markov Chains
[*] Regular Variation
[*] Asymptotic Properties of Regularly Varying Functions
[*] Problems for Solution
[/LIST]
[*] Infinitely Divisible Distributions and Semi-Groups
[LIST]
[*] Orientation
[*] Convolution Semi-Groups
[*] Preparatory Lemmas
[*] Finite Variances
[*] The Main Theorems
[*] Example: Stable Semi-Groups
[*] Triangular Arrays with Identical Distributions
[*] Domains of Attraction
[*] Variable Distributions. The Three-Series Theorem
[*] Problems for Solution
[/LIST]
[*] Markov Processes and Semi-Groups
[LIST]
[*] The Pseudo-Poisson Type
[*] A Variant: Linear Increments
[*] Jump Processes
[*] Diffusion Processes in [itex]\mathbb{R}^1[/itex]
[*] The Forward Equation. Boundary Conditions
[*] Diffusion in Higher Dimensions
[*] Subordinated Processes
[*] Markov Processes and Semi-Groups
[*] The "Exponential Formula" of Semi-Group Theory
[*] Generators. The Backward Equation
[/LIST]
[*] Renewal Theory
[LIST]
[*] The Renewal Theorem
[*] Proof of the Renewal Theorem
[*] Refinements
[*] Persistent Renewal Processes
[*] The Number [itex]N_t[/itex] of Renewal Epochs
[*] Terminating (Transient) Processes
[*] Diverse Applications
[*] Existence of Limits in Stochastic Processes
[*] Renewal Theory on the Whole Line
[*] Problems for Solution
[/LIST]
[*] Random Walks in [itex]\mathcal{R}^1[/itex]
[LIST]
[*] Basic Concepts and Notations
[*] Duality. Types of Random Walks
[*] Distribution of Ladder Heights. Wiener-Hopf Factorization
[LIST]
[*] The Wiener-Hopf Integral Equation
[/LIST]
[*] Examples
[*] Applications
[*] A Combinatorial Lemma
[*] Distribution of Ladder Epochs
[*] The Arc Sine Laws
[*] Miscellaneous Complements
[*] Problems for Solution
[/LIST]
[*] Laplace Transforms. Tauberian Theorems. Resolvents
[LIST]
[*] Definitions. The Continuity Theorem
[*] Elementary Properties
[*] Examples
[*] Completely Monotone Functions. Inversion Formulas
[*] Tauberian Theorems
[*] Stable Distributions
[*] Infinitely Divisible Distributions
[*] Higher Dimensions
[*] Laplace Transforms for Semi-Groups
[*] The Hille-Yosida Theorem
[*] Problems for Solution
[/LIST]
[*] Applications of Laplace Transforms
[LIST]
[*] The Renewal Equation: Theory
[*] Renewal-Type Equations: Examples
[*] Limit Theorems Involving Arc Sine Distributions
[*] Busy Periods and Related Branching Processes
[*] Diffusion Processes
[*] Birth-and-Death Processes and Random Walks
[*] The Kolmogorov Differential Equations
[*] Example: The Pure Birth Process
[*] Calculation of Ergodic Limits and of First-Passage Times
[*] Problems for Solution
[/LIST]
[*] Characteristic Functions
[LIST]
[*] Definition. Basic Properties
[*] Special Distributions. Mixtures
[LIST]
[*] Some Unexpected Phenomena
[/LIST]
[*] Uniqueness. Inversion Formulas
[*] Regularity Properties
[*] The Central Limit Theorem for Equal Components
[*] The Lindeberg Conditions
[*] Characteristic Functions in Higher Dimensions
[*] Two Characterizations of the Normal Distribution
[*] Problems for Solution
[/LIST]
[*] Expansions Related to the Central Limit Theorem
[LIST]
[*] Notations
[*] Expansions for Densities
[*] Smoothing
[*] Expansions for Distributions
[*] The Berry-Esseen Theorems
[*] Expansions in the Case of Varying Components
[*] Large Deviations
[/LIST]
[*] Infinitely Divisible Distributions
[LIST]
[*] Infinitely Divisible Distributions 554
[*] Canonical Forms. The Main Limit Theorem
[LIST]
[*] Derivatives of Characteristic Functions
[/LIST]
[*] Examples and Special Properties
[*] Special Properties
[*] Stable Distributions and Their Domains of Attraction
[*] Stable Densities
[*] Triangular Arrays
[*] The Class L
[*] Partial Attraction. "Universal Laws"
[*] Infinite Convolutions
[*] Higher Dimensions
[*] Problems for Solution
[/LIST]
[*] Applications of Fourier Methods to Random Walks
[LIST]
[*] The Basic Identity
[*] Finite Intervals. Wald's Approximation
[*] The Wiener-Hopf Factorization
[*] Implications and Applications
[*] Two Deeper Theorems
[*] Criteria for Persistency
[*] Problems for Solution
[/LIST]
[*] Harmonic Analysis
[LIST]
[*] The Parseval Relation
[*] Positive Definite Functions
[*] Stationary Processes
[*] Fourier Series
[*] The Poisson Summation Formula
[*] Positive Definite Sequences
[*] [itex]L^2[/itex] Theory
[*] Stochastic Processes and Integrals
[*] Problems for Solution
[/LIST]
[*] Answers to Problems
[*] Some Books on Cognate Subjects
[*] Index
[/LIST]
```

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