Contents |
Distribution Fitting Software

# Distribution Fitting — Reliability Analysis

## Introduction

## Problem

## Solution

### Exponential Distribution

()

### Weibull Distribution

()

### Lognormal Distribution

()

## Summary

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Reliability modelling based on lifetime distributions is probably one
of the most widely used reliability engineering techniques. This
technique allows to predict the reliability of a component or system, or
*the probability that a component or system will perform its required
function during a specified period of time under stated conditions*.

A company produces standard electronic circuits used to build more complex equipment. The goal is to perform reliability modelling of considered circuits using a data set of 50 observed survival times. Specifically,

- develop an appropriate survival model, and
- define the failure rate (hazard function).

To develop a valid model, we need to choose an appropriate probability distribution which can be used for modelling survival data. Since survival times are non-negative by nature, it makes sence to use probability distributions having a non-negative range of definition. In reliability analysis, several probability distributions are commony used. Among them are the Exponential, Weibull, and Lognormal distributions. The Gamma and Extreme Value (Gumbel) lifetime distributions are also sometimes applied.

This distribution is usually applied to model systems with constant failure rate. Its cumulative distribution function (CDF) is defined as follows:

The distribution parameter
(*lambda*) is a *failure rate* of a component or a system under
consideration. The two-parameter Exponential distribution, also known
as the shifted Exponential distribution, is sometimes used as well:

In this model, the location parameter
(*gamma*) represents a *waiting time* of a component or system.

The Weibull distribution is probably one of the most widely used lifetime distributions in reliability engineering. The two-parameter Weibull distribution is defined by the following cumulative distribution function:

The three-parameter Weibull is a more general distribution which also has
a *waiting time* parameter
(*gamma*):

The Lognormal distribution is a flexible distribution generally used to model failures caused by degradation processes (such as corrosion and fatigue), failure times of electronic units etc. The Lognormal CDF is expressed as

**Note:** It is very important to select the *best fitting*
survival distribution, because the model it represents will be used to
make key decisions. For example, you cannot assume the Exponential
distribution just because it is the simplest one - in fact, it is
inappropriate in many cases. The use of incorrect models can lead to
serious problems such as damage of expensive equipment, premature
failures of products resulting in unsatisfied customers etc.

The best way to prevent possible modelling errors, develop more valid models, and thus make better decisions, is to apply distribution fitting. This technique allows to select the probability distribution which best describes the reliability of a component or system, based on available historical data (observed survival times). However, the use of distribution fitting is connected with complex calculations which require special knowledge in the field of statistics and/or programming skills.

The problem of selecting the best fitting distribution can be easily solved by applying the specialized distribution fitting software EasyFit. This software product is designed to automate the whole distribution fitting process. It performs all the calculations for you, so you just need to interpret the analysis results and actually select the best model.

In this example, we use the Exponential, two-parameter Weibull, and Lognormal distributions. Our data set consists of the survival times (in 1000 hours) for 50 electronic circuits. Based on these data, EasyFit automatically estimates parameters of the chosen distributions:

We can apply the goodness of fit tests to compare the fitted distributions:

Both Kolmogorov-Smirnov and Anderson-Darling tests suggest that the
Weibull distribuion with parameters *alpha=1.15629* and *beta=58.34934*
fits to our data in the best way. It is also obvious that the Exponential
distribution fits relatively poorly.

The survival function graph allows us to visually compare the fitted Weibull reliability distribution with the empirical survival function:

The survival function shows that about 20% of our electronic circuits will survive beyond 10 years (87,600 hours) of operation: S(87.6) = 0.202.

Also, we can use the constructed survival model to determine the hazard function showing the instantaneous failure rate at any point in time:

For example, for X=4 years (35,040 hours) of operation, the estimated instantaneous failure rate is h(35.04) = 0.0183, or 1.83%.

We used EasyFit to develop a survival model of electronic circuits by analyzing available survival data. We fitted the most commonly used survival distributions, and found that the Weibull distribution fits to our data in the best way. Finally, we used the developed model to define the survival function and the hazard function.