Probability and Stochastic Processes - Ionut Florescu.pdf

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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Acknowledgments
Introduction
Part I: Probability
Chapter 1: Elements of Probability
Measure
1.1 Probability Spaces
1.2 Conditional Probability
1.3 Independence
1.4 Monotone Convergence
Properties of Probability
1.5 Lebesgue Measure on the Unit
Interval (0,1]
Problems
Chapter 2: Random Variables
Reduction to
. Random variables
2.1 Discrete and Continuous Random
Variables
2.2 Examples of Commonly
Encountered Random Variables
2
2.3 Existence of Random Variables
with Prescribed Distribution.
Skorohod Representation of a
Random Variable
2.4 Independence
2.5 Functions of Random Variables.
Calculating Distributions
Problems
Chapter 3: Applied Chapter: Generating
Random Variables
3.1 Generating One-Dimensional
Random Variables by Inverting the
cdf
3.2 Generating One-Dimensional
Normal Random Variables
3.3 Generating Random Variables.
Rejection Sampling Method
3.4 Generating Random Variables.
Importance Sampling
Problems
Chapter 4: Integration Theory
4.1 Integral of Measurable Functions
4.2 Expectations
4.3 Moments of a Random Variable.
Variance and the Correlation
Coefficient
4.4 Functions of Random Variables.
The Transport Formula
3
4.5 Applications. Exercises in
Probability Reasoning
4.6 A Basic Central Limit Theorem:
The DeMoivre–LaplaceTheorem:
Problems
Chapter 5: Product Spaces. Conditional
Distribution and Conditional
Expectation
5.1 Product Spaces
5.2 Conditional Distribution and
Expectation. Calculation in Simple
Cases
5.3 Conditional Expectation. General
Definition
5.4 Random Vectors. Moments and
Distributions
Problems
Chapter 6: Tools to study Probability.
Generating Function, Moment
Generating Function, Characteristic
Function
6.1 Sums of Random Variables.
Convolutions
6.2 Generating Functions and
Applications
6.3 Moment Generating Function
6.4 Characteristic Function
6.5 Inversion and Continuity
Theorems
4
6.6 Stable Distributions. Lévy
Distribution
Problems
Chapter 7: Limit Theorems
Introduction
7.1 Types of Convergence
7.2 Relationships between Types of
Convergence
7.3 Continuous Mapping Theorem.
Joint Convergence. Slutsky's
Theorem
7.4 The Two Big Limit Theorems:
LLN and CLT
7.5 Extensions of Central Limit
Theorem. Limit Theorems for Other
Types of Statistics
7.6 Exchanging the Order of Limits
and Expectations
Problems
Chapter 8: Statistical Inference
8.1 The Classical Problems in
Statistics
8.2 Parameter Estimation Problem
8.3 Maximum Likelihood Estimation
Method
8.4 The Method of Moments
8.5 Testing, the Likelihood Ratio Test
8.6 Confidence Sets
5

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