The Practical Guide
To Truncated Probability Distributions
Author: Claudio San Roman Denegri
First Edition September 2020
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Introductory Videos: Video 1 Video 2 Brief Example: real case

CONTENTS 



Page 

Preface 
8 



I 
Description of Truncated Distributions 
11 



1.1 
Definition 
12 
1.2 
Properties 
12 
1.3 
Use of Decibels to Graph Probability Curves 
13 
1.4 
Example 
13 
1.5 
Industry Context 
16 
1.6 
Notes about Censored Data 
18 



II 
Analysis of Truncated Distributions 
19 



2.1 
The Mean 
21 
2.2 
The Median 
22 
2.3 
The Mode 
23 
2.4 
The Variance 
24 
2.5 
The Asymmetry 
25 
2.6 
The Kurtosis 
27 



III 
Parameter Estimation of Truncated Distributions 
29 



3.1 
Engineering Methods 
30 
3.1.1 
The Artifice Mirror with Mode 
32 
3.1.2 
The Artifice Optimal Mirror 
35 
3.1.3 
Test of the Artifice Optimal Mirror 
40 
3.1.4 
Techniques for Double Truncated Distributions 
47 
3.1.4.1 
Estimating the Tail of the Generalized Gaussian Distribution using Polynomials 
48 
3.1.4.2 
Estimating β of the Generalized Gaussian Distribution using Shape of Head 
49 
3.1.4.3 
Double Truncated Distribution Example 
50 



3.2 
“Pure” Mathematic Methods 
54 
3.2.1 
The Truncated Generalized Gaussian 
54 
3.2.2 
The Double Truncated Generalized Gaussian 
54 



IV 
Appendix 
55 



A1 
The Generalized Gaussian Distribution 
55 
A2 
NonTruncated Distribution Methods to Estimate Parameters 
58 

A2.1 Moment Method 
59 

A2.2 Maximum Likelihood Estimation Method 
60 

A2.3 The Global Convergence Method 
62 

A2.4 Minimization of The Squared Error 
63 



V 
References 
64 


PREFACE
This book contains practical knowledge about truncated distributions. It applies mainly to symmetrical distributions, with special emphasis on the Truncated Generalized Gaussian Distribution. This book and the “pure” mathematical books related to truncated distributions complement each other.
Truncated distributions were complex subjects before. Now, with the knowledge provided by this book, these discontinuous functions are easy to analyze and solve. This book should be read by students and professionals that deal with statistics like economists, biologists, engineers, mathematicians, doctors, sociologists, astronomers, agronomists, psychologists, educators, etc.
To understand this book, it is required to complete university statistic courses.
The first chapter is about the description of truncated distributions, the reader will learn what the truncated distributions are. Analysis of truncated distribution are made in the second chapter, it contains information to understand how parameters change with different levels of truncation. The third chapter is dedicated entirely on how to estimate parameters, and the test of these tools. Also, there is a section about double truncated distributions. The appendix is located at the fourth section, it provides important tools to understand the first three chapters. And finally, the references are included in the fifth section.
A part of this book was written during the difficult days of coronavirus, I am glad that I could finish the first edition and share it with you. I wish the entire community a good health.

Review 1:
I read this book in a couple of weeks and I found it very interesting. This book has a lot of graphs and practical math, instead of large and complex equations. I learned that the methods presented in this book enable many conventional methods of nontruncated distributions to be applied to truncated distributions, I believe this is a major contribution to mathematics. I strongly recommend this book.
Andres. November 2020
Review 2:
This book showed me the idea of using decibels (or logarithmic scale) to work with probability curves, it was for me a new idea, I could better understand the tails of distributions and compare with other curves. I also liked the use of Monte Carlo method to analyze truncated distributions and to test methods. This book is a key source for students and professionals.
Gabriel. December 2020
Review 3:
Now I understand. When you have data limited by time, capacity, direction, sensorrange or others, you get truncated curves. In my opinion, this book gives excellent tools for this topic.
Aristides. February 2021
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