Probability Distributions Used in Reliability Engineering

1 Probability Distributions Used in Reliability Engineering Probability Distributions Used in Reliability Engineering Andrew N. O'Connor Mohammad Modarres Ali Mosleh Center for Risk and Reliability 0151 Glenn L Martin Hall University of Maryland College Park, Maryland Published by the Center for Risk and Reliability International Standard Book Number (ISBN): 978-0-9966468-1-9. Copyright 2016 by the Center for Reliability Engineering University of Maryland, College Park, Maryland, USA. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from The Center for Reliability Engineering , Reliability Engineering Program.

2 The Center for Risk and Reliability University of Maryland College Park, Maryland 20742-7531. In memory of Willie Mae Webb This book is dedicated to the memory of Miss Willie Webb who passed away on April 10 2007 while working at the Center for Risk and Reliability at the University of Maryland (UMD). She initiated the concept of this book, as an aid for students conducting studies in Reliability Engineering at the University of Maryland. Upon passing, Willie bequeathed her belongings to fund a scholarship providing financial support to Reliability Engineering students at UMD. Preface Reliability Engineers are required to combine a practical understanding of science and Engineering with statistics. The Reliability engineer's understanding of statistics is focused on the practical application of a wide variety of accepted statistical methods.

3 Most Reliability texts provide only a basic introduction to Probability Distributions or only provide a detailed reference to few Distributions . Most texts in statistics provide theoretical detail which is outside the scope of likely Reliability Engineering tasks. As such the objective of this book is to provide a single reference text of closed form Probability formulas and approximations used in Reliability Engineering . This book provides details on 22 Probability Distributions . Each distribution section provides a graphical visualization and formulas for distribution parameters, along with distribution formulas. Common statistics such as moments and percentile formulas are followed by likelihood functions and in many cases the derivation of maximum likelihood estimates.

4 Bayesian non-informative and conjugate priors are provided followed by a discussion on the distribution characteristics and applications in Reliability Engineering . Each section is concluded with online and hardcopy references which can provide further information followed by the relationship to other Distributions . The book is divided into six parts. Part 1 provides a brief coverage of the fundamentals of Probability Distributions within a Reliability Engineering context. Part 1 is limited to concise explanations aimed to familiarize readers. For further understanding the reader is referred to the references. Part 2 to Part 6 cover Common Life Distributions , Univariate Continuous Distributions , Univariate Discrete Distributions and Multivariate Distributions respectively.

5 The authors would like to thank the many students in the Reliability Engineering Program particularly Reuel Smith for proof reading. Contents i Contents PREFACE .. V. CONTENTS .. I. 1. FUNDAMENTALS OF Probability Distributions .. 1. Probability Theory .. 2. Theory of Probability .. 2. Interpretations of Probability Theory .. 2. Laws of 3. Law of Total Probability .. 4. Bayes' Law .. 4. Likelihood Functions .. 5. Fisher Information Matrix .. 6. Distribution Functions .. 9. Random Variables .. 9. Statistical Distribution Parameters .. 9. Probability Density Function .. 9. Cumulative Distribution Function .. 11. Reliability Function .. 12. Conditional Reliability Function .. 13. 100 % Percentile Function .. 13. Mean Residual 13. Hazard Rate .. 13. Cumulative Hazard Rate.

6 14. Characteristic Function .. 15. Joint Distributions .. 16. Marginal Distribution .. 17. Conditional 17. Bathtub 17. Truncated Distributions .. 18. Summary .. 19. Distribution Properties .. 20. Median / Mode .. 20. Moments of Distribution .. 20. Covariance .. 21. Parameter Estimation .. 22. Probability Plotting Paper .. 22. Total Time on Test Plots .. 23. Least Mean Square Regression .. 24. Method of Moments .. 25. Maximum Likelihood Estimates .. 26. Bayesian Estimation .. 27. ii Probability Distributions Used in Reliability Engineering Confidence Intervals .. 30. Related Distributions .. 33. Supporting Functions .. 34. Beta Function , .. 34. Incomplete Beta Function ( ; , ) .. 34. Regularized Incomplete Beta Function ( ; , ) .. 34. Complete Gamma Function ( ).

7 34. Upper Incomplete Gamma Function ( , ) .. 35. Lower Incomplete Gamma Function ( , ).. 35. Digamma Function .. 36. Trigamma Function .. 36. Referred Distributions .. 37. Inverse Gamma Distribution ( , ).. 37. Student T Distribution ( , , ) .. 37. F Distribution ( , ) .. 37. Chi-Square Distribution ( ) .. 37. Hypergeometric Distribution ( ; , , ) .. 38. Wishart Distribution ( ; , ) .. 38. Nomenclature and Notation .. 39. 2. COMMON LIFE Distributions ..40. Exponential Continuous Distribution .. 41. Lognormal Continuous Distribution .. 49. Weibull Continuous Distribution .. 59. 3. BATHTUB LIFE Distributions ..68. 2-Fold Mixed Weibull Distribution .. 69. Exponentiated Weibull Distribution .. 76. Modified Weibull Distribution .. 81. 4. UNIVARIATE CONTINUOUS Distributions .

8 85. Beta Continuous Distribution .. 86. Birnbaum Saunders Continuous Distribution .. 93. Contents iii Gamma Continuous Distribution .. 99. Logistic Continuous Distribution .. 108. Normal (Gaussian) Continuous Distribution .. 115. Pareto Continuous Distribution .. 125. Triangle Continuous Distribution .. 131. Truncated Normal Continuous 135. Uniform Continuous Distribution .. 145. 5. UNIVARIATE DISCRETE Distributions .. 151. Bernoulli Discrete Distribution .. 152. Binomial Discrete Distribution .. 157. Poisson Discrete Distribution .. 165. 6. BIVARIATE AND MULTIVARIATE Distributions .. 172. Bivariate Normal Continuous Distribution .. 173. Dirichlet Continuous Distribution .. 181. Multivariate Normal Continuous Distribution .. 187. Multinomial Discrete Distribution.

9 193. 7. REFERENCES .. 200. iv Probability Distributions Used in Reliability Engineering Introduction 1. Prob Theory 1. Fundamentals of Probability Distributions 2 Probability Distributions Used in Reliability Engineering Prob Theory Probability Theory Theory of Probability The theory of Probability formalizes the representation of probabilistic concepts through a set of rules. The most common reference to formalizing the rules of Probability is through a set of axioms proposed by Kolmogorov in 1933. Where is an event in the event space = =1 with different events. 0 ( ) 1. ( ) = 1 and ( ) = 0. ( 1 2 ) = ( 1 ) + ( 2 ). When 1 and 2 are mutually exclusive. Other representations of uncertainty exist such as fuzzy logic and theory of evidence (Dempster-Shafer model) which do not follow the theory of Probability but almost all Reliability concepts are defined based on Probability as the metric of uncertainty.

10 For a justification of Probability theory see (Singpurwalla 2006). Interpretations of Probability The two most common interpretations of Probability are: Frequency Interpretation. In the frequentist interpretation of Probability , the Probability of an event (failure) is defined as: . ( ) = lim . Also known as the classical approach, this interpretation assumes there exists a real Probability of an event, . The analyst uses the observed frequency of the event to estimate the value of . The more historic events that have occurred, the more confident the analyst is of the estimation of . This approach does have limitations, for instance when data from events are not available ( no failures occur in a test) cannot be estimated and this method cannot incorporate soft evidence such as expert opinion.