Probability Distributions Used in Reliability Engineering

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.

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. 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. 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.

The authors would like to thank the many students in the Reliability Engineering Program particularly Reuel Smith for proof reading.

Contents

Contents i

PREFACE ......................................................................................................................................................V

CONTENTS .................................................................................................................................................... I

1. FUNDAMENTALS OF PROBABILITY DISTRIBUTIONS...................................................... 1

1.1. Probability Theory .......................................................................................................... 2 1.1.1. Theory of Probability ................................................................................................ 2 1.1.2. Interpretations of Probability Theory ....................................................................... 2 1.1.3. Laws of Probability.................................................................................................... 3 1.1.4. Law of Total Probability ............................................................................................ 4 1.1.5. Bayes' Law ................................................................................................................ 4 1.1.6. Likelihood Functions ................................................................................................. 5 1.1.7. Fisher Information Matrix......................................................................................... 6

1.2. Distribution Functions .................................................................................................... 9 1.2.1. Random Variables..................................................................................................... 9 1.2.2. Statistical Distribution Parameters ........................................................................... 9 1.2.3. Probability Density Function..................................................................................... 9 1.2.4. Cumulative Distribution Function ........................................................................... 11 1.2.5. Reliability Function ................................................................................................. 12 1.2.6. Conditional Reliability Function .............................................................................. 13 1.2.7. 100% Percentile Function ..................................................................................... 13 1.2.8. Mean Residual Life.................................................................................................. 13 1.2.9. Hazard Rate ............................................................................................................ 13 1.2.10. Cumulative Hazard Rate ......................................................................................... 14 1.2.11. Characteristic Function ........................................................................................... 15 1.2.12. Joint Distributions................................................................................................... 16 1.2.13. Marginal Distribution.............................................................................................. 17 1.2.14. Conditional Distribution.......................................................................................... 17 1.2.15. Bathtub Distributions.............................................................................................. 17 1.2.16. Truncated Distributions .......................................................................................... 18 1.2.17. Summary................................................................................................................. 19

1.3. Distribution Properties ................................................................................................. 20 1.3.1. Median / Mode....................................................................................................... 20 1.3.2. Moments of Distribution ........................................................................................ 20 1.3.3. Covariance .............................................................................................................. 21

1.4. Parameter Estimation ................................................................................................... 22 1.4.1. Probability Plotting Paper ....................................................................................... 22 1.4.2. Total Time on Test Plots ......................................................................................... 23 1.4.3. Least Mean Square Regression ............................................................................... 24 1.4.4. Method of Moments .............................................................................................. 25 1.4.5. Maximum Likelihood Estimates .............................................................................. 26 1.4.6. Bayesian Estimation................................................................................................ 27

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Probability Distributions Used in Reliability Engineering

1.4.7. Confidence Intervals ............................................................................................... 30

1.5. Related Distributions .................................................................................................... 33

1.6. Supporting Functions.................................................................................................... 34 1.6.1. Beta Function , ................................................................................................ 34 1.6.2. Incomplete Beta Function (; , ) .................................................................... 34 1.6.3. Regularized Incomplete Beta Function (; , ).................................................. 34 1.6.4. Complete Gamma Function () ........................................................................... 34 1.6.5. Upper Incomplete Gamma Function (, ) .......................................................... 35 1.6.6. Lower Incomplete Gamma Function (, )........................................................... 35 1.6.7. Digamma Function ........................................................................................... 36 1.6.8. Trigamma Function .......................................................................................... 36

1.7. Referred Distributions .................................................................................................. 37 1.7.1. Inverse Gamma Distribution (, ).................................................................... 37 1.7.2. Student T Distribution (, , ) ......................................................................... 37 1.7.3. F Distribution (, ) ........................................................................................ 37 1.7.4. Chi-Square Distribution () ............................................................................... 37 1.7.5. Hypergeometric Distribution (; , , )....................................... 38 1.7.6. Wishart Distribution (; , ) .............................................................. 38

1.8. Nomenclature and Notation......................................................................................... 39

2. COMMON LIFE DISTRIBUTIONS ............................................................................................40

2.1. Exponential Continuous Distribution............................................................................ 41

2.2. Lognormal Continuous Distribution.............................................................................. 49

2.3. Weibull Continuous Distribution .................................................................................. 59

3. BATHTUB LIFE DISTRIBUTIONS ...........................................................................................68

3.1. 2-Fold Mixed Weibull Distribution................................................................................ 69

3.2. Exponentiated Weibull Distribution ............................................................................. 76

3.3. Modified Weibull Distribution ...................................................................................... 81

4. UNIVARIATE CONTINUOUS DISTRIBUTIONS ...................................................................85

4.1. Beta Continuous Distribution ....................................................................................... 86

4.2. Birnbaum Saunders Continuous Distribution ............................................................... 93

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