Probability Distributions: With Truncated, Log and Bivariate Extensions

ISBN-13: 9783030093884 / Angielski / Miękka / 2019 / 163 str.

Nick T. Thomopoulos

Recenzje

"The book is clearly written, and makes it easy to read and comprehend the various issues related to probability distributions." (Sada Nand Dwivedi, ISCB News, iscb.info, Issue 66, December, 2018)
"The book gives a concise and practical overview of the commonly used distributions and statistical methods not presented in other publications. Table values and examples are provided as well. ... The book is free of theorems. It is intended for anyone who pursues statistical and probability analysis." (Oleg K. Zakusilo, zbMATH 1396.60003, 2018)

Spis treści

1. Continuous Distributions

1.1 Introduction

1.2 Sample Data Statistics

1.3 Notation

1.4 Parameter Estimating Methods

1.5 Transforming Variables

Transform Data to (0,1)

Transform Data to (x ≥0)

1.6 Continuous Random Variables

1.7 Continuous Uniform 

Coefficient of Variation

Parameter Estimates

1.8 Exponential 

1.9 Erlang  

Parameter Estimates

1.10 Gamma  

Parameter Estimates

1.11 Beta  

Standard Beta  

Mean and Variance 

Parameter Estimates

1.12    Weibull  

Weibull Plot 

Parameter Estimates

1.13 Normal

Standard Normal Distribution 

Coefficient of Variation

Parameter Estimates

1.14 Lognormal     

Parameter Estimates

1.15 Summary

1.16   Reference

2   Discrete Distributions

2.1   Introduction

2.2   Discrete Random Variables

Lexis Ratio

2.3   Discrete Uniform

Parameter Estimates

2.4   Binomial

Lexis Ratio

Parameter Estimates

Normal Approximation  

Poisson Approximation  

2.5 Geometric

Number of Trials

Number of Failures

Lexis Ratio

Parameter Estimate

2.6   Pascal

Number of Trials

Lexis Ratio 

Parameter Estimate

Number of Failures

Lexis Ratio

Parameter Estimate

2.7   Poisson

Lexis Ratio

Relation to the Exponential Distribution

Parameter Estimate

2.8   Hyper Geometric

Parameter Estimate2.9 Summary

2.10   Reference

3 Standard Normal

3.1 Introduction

3.2 Gaussian Distribution

3.3 Some Relations on the Standard Normal Distribution

4.3 Normal Distribution

3.5 Standard Normal

3.6     Hastings Approximations

Approximation of F(z) from z

Approximation of z from F(z)

3.7 Table Values of the Standard Normal

3.8 Discrete Normal Distribution

3.9 Summary

3.10   References

4 Partial Expectation

4.1 Introduction

4.2 Partial Expectation

4.3 Left Location Parameter

Table Entries

4.4 Inventory Management

4.5 Right Location Parameter

4.6 Advance Demand

4.7   Summary

4.8   References

5 Left Truncated Normal

5.1   Introduction

5.2   Left-Location Parameter

5.3   Mathematical Equations

5.4   Table Entries

5.5   More Tables

5.6   Left Truncated Distribution

5.7 Application to Sample Data

5.8   LTN for Inventory Control

Automotive Service Parts Distribution Center

Retail Products

5.9   Summary

5.10   References

6   Right Truncated Normal

6.1   Introduction

6.2   Right Truncated Distribution

6.3   Mathematical Equations

6.4   Variable t Range

6.5 Table Entries

6.6 Application to Sample Data

6.7   More Tables

6.8   Summary

6.9   Reference

7 Truncated Normal Spread Ratio

7.1   Introduction

7.2   The Spread Ratio

7.3   LTN Distribution Measures7.4   LTN Table Entries

7.5   RTN Distribution Measures

7.6   RTN Table Entries

7.7   Estimating the Distribution Type

7.8   Selecting the Distribution Type

7.9 Estimating the Low and High Limits

When LTN

Estimate  When LTN

When RTN

Estimate When RTN

When Normal

Compute the Adjusted Coefficient of Variation

7.10   Find xwhere P(x ≤ x) = 

7.11   Find  where P(x ≤ x`) = 

7.12   Summary    

8   Bivariate Normal 

8.1   Introduction 

8.2   Bivariate Normal Distribution

Marginal Distributions

Conditional Distributions

8.3   Bivariate Standard Normal Distribution

Conditional Distribution of z2

Conditional Distribution of z1

Cumulative Joint Probability

Approximation of F(k1,k2)

Table Values of F(k1,k2) 

8.4   Some Basic Probabilities for (z1,z2) ~ BVN(0,0,1,1,)

8.5   Probabilities for (x1,x2) ~ BVN

8.6   Summary

8.7   References

9     Lognormal

9.1 Introduction

9.2   Lognormal Distribution

9.3   Notation

9.4   Lognormal

Lognormal Mode 

Lognormal Median

9.5   Raw Lognormal Variable

9.6   Shifted Lognormal Variable

9.7   Normal Variable

9.8 Zero-Mean Normal Variable

9.9 Standard LN Variable

9.10 Lognormal Table Entries

9.11 Lognormal Distribution Table

9.12   Summary

9.13   Reference

10 Bivariate Lognormal 

10.1   Introduction

10.2   Bivariate Lognormal      

Notation

Some Properties Between x and y

Mode of x and x`

10.3   Lognormal and Normal Notation

Related Parameters

10.4   Bivariate Lognormal Distribution

Bivariate Lognormal Correlation

Bivariate Lognormal Designation

10.5   Bivariate Normal Distribution

10.6   Bivariate (Zero-Mean) Normal Distribution

10.7   Bivariate (Standard) Normal Distribution

10.8   Summary

10.9   References

O autorze

Nick T. Thomopoulos, Ph.D., has degrees in business (B.S.) and in mathematics (M.A.) from the University of Illinois, and in industrial engineering (Ph.D.) from Illinois Institute of Technology (Illinois Tech). He was supervisor of operations research at International Harvester; senior scientist at the IIT Research Institute; and Professor in Industrial Engineering and in the Stuart School of Business at Illinois Tech. He is the author of eleven books including Fundamentals of Queuing Systems (Springer), Essentials of Monte Carlo Simulation (Springer), Applied Forecasting Methods (Prentice Hall), and Fundamentals of Production, Inventory and the Supply Chain (Atlantic). He has published many papers and has consulted in a wide variety of industries in the United States, Europe and Asia. Dr. Thomopoulos has received honors over the years, such as the Rist Prize from the Military Operations Research Society for new developments in queuing theory; the Distinguished Professor Award in Bangkok, Thailand from the Illinois Tech Asian Alumni Association; and the Professional Achievement Award from the Illinois Tech Alumni Association. 

Blurb

This volume presents a concise and practical overview of statistical methods and tables not readily available in other publications.  It begins with a review of the commonly used continuous and discrete probability distributions. Several useful distributions that are not so common and less understood are described with examples and applications in full detail: discrete normal, left-partial, right-partial, left-truncated normal, right-truncated normal, lognormal, bivariate normal, and bivariate lognormal. Table values are provided with examples that enable researchers to easily apply the distributions to real applications and sample data. The left- and right-truncated normal distributions offer a wide variety of shapes in contrast to the symmetrically shaped normal distribution, and a newly developed spread ratio enables analysts to determine which of the three distributions best fits a particular set of sample data.  The book will be highly useful to anyone who does statistical and probability analysis. This includes scientists, economists, management scientists, market researchers, engineers, mathematicians, and students in many disciplines.

Nick T. Thomopoulos, Ph.D., has degrees in business (B.S.) and in mathematics (M.A.) from the University of Illinois, and in industrial engineering (Ph.D.) from Illinois Institute of Technology (Illinois Tech). He was supervisor of operations research at International Harvester; senior scientist at the IIT Research Institute; and Professor in Industrial Engineering and in the Stuart School of Business at Illinois Tech. He is the author of eleven books including Fundamentals of Queuing Systems (Springer), Essentials of Monte Carlo Simulation (Springer), Applied Forecasting Methods (Prentice Hall), and Fundamentals of Production, Inventory and the Supply Chain (Atlantic). He has published many papers and has consulted in a wide variety of industries in the United States, Europe and Asia. Dr. Thomopoulos has received honors over the years, such as the Rist Prize from the Military Operations Research Society for new developments in queuing theory; the Distinguished Professor Award in Bangkok, Thailand from the Illinois Tech Asian Alumni Association; and the Professional Achievement Award from the Illinois Tech Alumni Association.