Statistical Analysis of Reliability and Life-Testing Models

Theory and Methods, Second Edition,

Lee Bain

• Publisher: Routledge
• Year: 2017
• eBook ISBN: 9781351414647
• Edition: 2
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Textbook for a methods course or reference for an experimenter who is mainly interested in data analyses rather than in the mathematical development of the procedures. Provides the most useful statistical techniques, not only for the normal distribution, but for other important distributions, such a

STATISTICAL ANALYSIS OF RELIABILITY AND LIFE-TESTING MODELS

1

Probabilistic Models

1. PROBABILITY

1.1. Introduction

Suppose a certain physical phenomenon is of interest, and an experiment is conducted to obtain an observed value of this phenomenon. It may be possible to develop a mathematical model so that if the conditions of the experiment are known, then the outcome can be predetermined, at least to a sufficient degree of accuracy. For example, Ohm’s law, E = IR, predicts the value of the electromotive force for a given level of current and resistance. Ohm’s law is an example of a deterministic mathematical model. However, it may be that the outcome of the experiment cannot be determined on the basis of the available knowledge of the experiment. In this case a probabilistic mathematical model is necessary.

There are many different reasons why a probabilistic model may be required. In some cases the outcomes may truly occur by chance, such as when a die is rolled, any of the six possible faces may occur, on the other hand, it may be that some of the conditions of the experiment are simply unknown or cannot be controlled. For example, two light bulbs may be manufactured by the same process and used under the same general conditions but still fail at different times. Similarly, a deterministic model may exist but be too complex to develop, or some factor such as measurement error may necessitate the use of a probabilistic model.

The purpose of this chapter is to set up the mathematical structure for describing a physical phenomenon in terms of a probability model. Chapter 2 will review some of the general statistical procedures which are helpful in analyzing and selecting an appropriate probability distribution to fit a particular problem of interest. Specific statistical techniques are then studied in detail in the following chapters for the most common probability models in use.

1.2. Notation

Again suppose some physical phenomenon is of interest, such as some characteristic of an item in a large population. This population may be either real or conceptual. For example, if the characteristic of interest is the length of time a light bulb made by a certain process will function properly, then there may be a large number of light bulbs already made, or, conceptually, there could be a large number made. Of course, the goal is to gain knowledge about th...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• 1. PROBABILISTIC MODELS
• 2. BASIC STATISTICAL INFERENCE
• 3. THE EXPONENTIAL DISTRIBUTION
• 4. THE WEIBULL DISTRIBUTION
• 5. THE GAMMA DISTRIBUTION
• 6. EXTREME-VALUE DISTRIBUTIONS
• 7. THE LOGISTIC AND OTHER DISTRIBUTIONS
• 8. GOODNESS-OF-FIT TESTS
• 9. REPAIRABLE SYSTEMS
• APPENDIXES
• REFERENCES
• INDEX