Most Commonly Used Distributions

There are many different lifetime distributions that can be used to model reliability data. Leemis [22] and others present a good overview of many of these distributions. In this reference, we will concentrate on the most commonly used and most widely applicable distributions for life data analysis, as outlined in the following sections.

The Weibull Distribution

The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the three-parameter Weibull pdf is defined by:

with three parameters β, η and γ, where β = shape parameter, η = scale parameter and γ = location parameter.

If the location parameter, γ, is assumed to be zero, the distribution then becomes the two-parameter Weibull or:

One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, γ, is zero, and the shape parameter is a known constant, or β = constant = C, so:

The Weibull Distribution chapter of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.

The Weibull-Bayesian Distribution

Another approach is the Weibull-Bayesian model which assumes that the analyst has some prior knowledge about the distribution of the shape parameter (β) of the Weibull distribution. There are many practical applications for this model, particularly when dealing with small sample sizes and/or some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data or physics-of-failure.

Note that this is not the same as the so called "WeiBayes" model. The so called "WeiBayes" model is really a one-parameter Weibull distribution. It assumes a fixed value (constant) for the shape parameter and solves for the scale parameter. The Weibull-Bayesian model in Weibull++ 7 is actually a true "WeiBayes" model and offers an alternative to the one-parameter Weibull by including the variation and uncertainty that is present in the prior estimation of the shape parameter.

The Weibull-Bayesian distribution and its characteristics are presented in more detail in the Weibull Distribution chapter.

The Exponential Distribution

The exponential distribution is commonly used for components or systems exhibiting a constant failure rate and is defined in its most general case by:

(8)

(also known as the two-parameter exponential in this form) with two parameters, namely λ and γ.

If the location parameter, γ, is assumed to be zero, the distribution then becomes the one-parameter exponential or,

The exponential distribution and its characteristics are presented in more detail in the Exponential Distribution chapter.

The Normal Distribution

The normal distribution is commonly used for general reliability analysis, times-to-failure of simple electronic and mechanical components, equipment or systems.

The pdf of the normal distribution is given by:

(9)

where,

μ = mean of the normal times to failure
σ = standard deviation of the times to failure

The normal distribution and its characteristics are presented in more detail in the Normal Distribution chapter.

The Lognormal Distribution

The lognormal distribution is commonly used for general reliability analysis, cycles-to-failure in fatigue, material strengths and loading variables in probabilistic design.

When the natural logarithms of the times-to-failure are normally distributed, then we say that the data follow the lognormal distribution.

The pdf of the lognormal distribution is given by:

(10)

where,

μ' = mean of the natural logarithms of the times-to-failure
σT' = standard deviation of the natural logarithms of the times to failure

The lognormal distribution and its characteristics are presented in more detail in the Lognormal Distribution chapter.