Bernoulli scheme in probability theory / Mohammad Ahsaah.

Format:

Book

Author/Creator:

Ahsaah, Mohammad, author.

Series:

Mathematics research developments series.

Mathematics Research Developments Series

Language:

English

Subjects (All):

Random variables.

Stochastic processes.

Physical Description:

1 online resource (188 pages)

Edition:

First edition.

Place of Publication:

New York : Nova Science Publishers, Inc., [2024]

Summary:

"The results of the probability theory help us to predict a lot of future events, to estimate our chances to win or to lose in many life situations. The beginning of this theory was connected with the investigations of the so-called Bernoulli random variables. These simplest variables take on only two values, but they describe many situations in our life. Up to now the different random schemes need to get and to use the new results for these two-valued random variables. The authors of this book want to give the readers useful information about the role of Bernoulli distributions in probability, in mathematical statistics and in their different applications"-- Provided by publisher.

Contents:

Intro

Contents

Preface

Introduction

Chapter 1

Elementary Probabilities

1.1. Random Events, Operations on Events, Probabilities, Probability Space

1.2. Probability Properties

1.3. The Classical Definition of Probabilities or the Scheme of Equiprobable Outcomes

1.4. Conditional Probabilities

1.4.1. Probability Multiplication Formula

1.5. Total Probability Formula. Bayes Formula

1.6. Independence of Events

Chapter 2

Random Variables

2.1. Random Variables and Their Distribution

2.2. The Distribution Function of a Random Variable and Its Properties

2.3. Various Types of Distribution of Random Variables

2.4. Random Vectors and Their Distribution

2.5. Independence of Random Variables

Chapter 3

Numerical Characteristics of Random Variables

3.1. Mathematical Expectation of a Random Variable

3.2. Dispersion (Degree of Scattering)

3.3. Moments of a Random Variable

Chapter 4

Definition of Bernoulli Tests: Random Variables with the Bernoulli Distribution Law

4.1. Bernoulli Random Variables

4.2. Bernoulli Trials

Chapter 5

Distributions Generated by the Bernoulli Distribution

5.1. Random Variables with Binomial Distribution

5.1.1. The Problem of the Most Probable Number of Successes

5.2. Random Variables with a Geometric Distribution Law

5.3. Limit Theorems in the Bernoulli Scheme. Normal Distribution

5.4. On the Connection between Bernoulli Trials and the Poisson Distribution

5.4.1. Poisson's Theorem

5.5. Negative Binomial Distribution

5.5.1. General Definition

5.6. Other Probability Distributions Associated with the Bernoulli Scheme

Chapter 6

Two-Valued Random Variables and Records

6.1. Definitions of Record Values

6.2. Record Indicators

6.3. Limit Theorems for Numbers of Records

6.4. Distributions of Record Times.

6.5. Moment Characteristics of Record Time

6.6. Solutions of the Problems

Chapter 7

Random Walk

7.1. Introduction

7.2. Symmetric Random Walk (i.e., p = q)

7.3. Random Walk-in Higher Dimension

7.4. Characteristic Function of a Simple Random Walk

Chapter 8

Records of Geometric and Exponential Distributions

8.1. Records of Discrete Distributions

8.2. Record Values of Geometric Distribution

8.3. Characterizations

8.4. Weak Records

8.5. Exponential Distribution

8.5.1. Distribution of Record Values

8.5.2. Moments of Record Values

8.5.3. Characterizations

References

Additional Reading

Appendix

A.1. Bernoulli Distribution

A.2. Binomial Distribution (B(n, p))

A.2.1. Introduction

A.2.2. Distributional Properties

A.3. Exponential Distribution (E(µ,σ))

A.3.1. Introduction

A.3.2. Distributional Properties

A.4. Geometric Distribution (GE(p))

A.4.1. Introduction

A.4.2. Distributional Properties

A.5. Hypergeometric Distribution (HG(a,b,n))

A.5.1. Introduction

A.5.2. Distributional Properties

A.6. Negative Binomial Distribution (NB(m, p))

A.6.1. Introduction

A.6.2. Distributional Properties

A.7. Normal Distribution (N(µ,σ))

A.7.1. Introduction

A.7.2. Central Limit Theorem

A.7.3. Distributional Properties

A.7.4. Order Statistics

A.8. Poisson Distribution (P())

A.8.1. Introduction

A.8.2. Distributional Properties

Index

Blank Page.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references and index.

ISBN:

9798895301357