Contents | Distribution Fitting Software

Featured Product
EasyFit - the easiest way to select the best fitting distribution and apply it to make better business decisions... learn more
Solve a wide range of business problems dealing with uncertainty using EasyFit as a stand-alone application or with Excel.
Download Free Trial
Free 30-Day Trial

Distribution Fitting — Reliability Analysis


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.

Exponential Distribution

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

Exponential CDF   ()

The distribution parameter lambda (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:

Shifted Exponential CDF

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

Weibull Distribution

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:

Two-parameter Weibull CDF   ()

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

Three-parameter Weibull CDF

Lognormal Distribution

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

Lognormal CDF   ()

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:

Estimated parameters

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

Goodness of fit tests

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:

Survival function graph

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:

Failure rate graph

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.