Distributions Used in Accelerated Testing

In this chapter, we will briefly present three lifetime distributions commonly used in accelerated life test analysis: the 1-parameter exponential, the 2-parameter Weibull and the lognormal distributions. Readers who are interested in a more rigorous overview (or in different forms of these and other life distributions) can refer to ReliaSoft's Life Data Analysis Reference. For information about the parameter estimation methods, see Appendix B.

=The Exponential Distribution=

The exponential distribution is a very commonly used distribution in reliability engineering. Due to its simplicity, it has been widely employed, even in cases where it doesn't apply. The exponential distribution is used to describe units that have a constant failure rate. The 1-parameter exponential pdf is given by:


 * $$\begin{align}

& f(T)= \lambda {{e}^{-\lambda T}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}T}} \\ & T\ge 0,\lambda >0,m>0 \end{align}$$

where:


 * λ = constant failure rate, in failures per unit of measurement (e.g. failures per hour, per cycle, etc.).
 * $$\lambda =\tfrac{1}{m}$$.
 * m = mean time between failures, or to a failure.
 * T = operating time, life, or age, in hours, cycles, miles, actuations, etc. This distribution requires the estimation of only one parameter, λ, for its application.

Characteristics of the Exponential Distribution
= The Weibull Distribution =

The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of β (slope). It can therefore model a great variety of data and life characteristics [18].

The 2-parameter Weibull pdf is given by:


 * $$f(T)=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}$$

where:


 * $$f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }$$

and:


 * η = scale parameter.
 * β = shape parameter (or slope).

Characteristics of the Weibull Distribution
= 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. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Since the logarithms of a lognormally distributed random variable are normally distributed, the lognormal distribution is given by:


 * $$f({T}')=\frac{1}{{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{T}'}}{{{\sigma }_}} \right)}^{2}}}}$$

where:


 * T' = ln T, and where the T s are the failure times.
 * $$\bar{{T}'}=$$ mean of the natural logarithms of the times to failure.
 * σT' = standard deviation of the natural logarithms of the failure times.

The lognormal pdf can be obtained, realizing that for equal probabilities under the normal and lognormal pdf s incremental areas should also be equal, or:


 * $$\begin{align}

f(T)dT = f(T')dT' \end{align}$$

Taking the derivative yields:


 * $$d{T}'=\frac{dT}{T}$$

Substitution yields:


 * $$\begin{align}

f(T)= \frac{f({T}')}{T}= \frac{1}{T\cdot {{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{T}'}}{{{\sigma }_}} \right)}^{2}}}} \end{align}$$

where:


 * $$f(T)\ge 0,T>0,-\infty <\bar{{T}'}<\infty ,{{\sigma }_{{{T}'}}}>0$$