Life Distributions

Distributions
A statistical distribution is fully described by its $$pdf$$ (or probability density function). In the previous sections, we used the definition of the $$pdf$$ to show how all other functions most commonly used in reliability engineering and life data analysis can be derived, namely the reliability function, failure rate function, mean time function and median life function, etc. All of these can be determined directly from the $$pdf$$ definition, or $$f(t)$$. Different distributions exist, such as the normal, exponential, etc., and each one of them has a predefined form of $$f(t).$$ These distribution definitions can be found in many references. In fact, entire texts have been dedicated to defining families of statistical distributions. These distributions were formulated by statisticians, mathematicians and engineers to mathematically model or represent certain behavior. For example, the Weibull distribution was formulated by Walloddi Weibull and thus it bears his name. Some distributions tend to better represent life data and are most commonly called lifetime distributions. One of the simplest and most commonly used distributions (and often erroneously overused due to its simplicity)', is the exponential distribution. The $$pdf$$ of the exponential distribution is mathematically defined as:


 * $$f(t)=\lambda e^{-\lambda t}$$

In this definition, note that $$t$$ is our random variable which represents time and the Greek letter $$\lambda $$ (lambda) represents what is commonly referred to as the parameter of the distribution. Depending on the value of $$\lambda ,$$  $$f(t)$$ will be scaled differently. For any distribution, the parameter or parameters of the distribution are estimated from the data. For example, the most well-known distribution, the normal (or Gaussian) distribution, is given by:


 * $$f(t)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}(\frac{t-\mu}{\sigma})^2}$$

$$\mu$$, the mean, and $$\sigma ,$$ the standard deviation, are its parameters. Both of these parameters are estimated from the data, i.e. the mean and standard deviation of the data. Once these parameters have been estimated, our function $$f(t)$$ is fully defined and we can obtain any value for $$f(t)$$ given any value of $$t$$. Given the mathematical representation of a distribution, we can also derive all of the functions needed for life data analysis, which again will depend only on the value of $$t$$ after the value of the distribution parameter or parameters have been estimated from data.undefined For example, we know that the exponential distribution $$pdf$$ is given by:


 * $$f(t)=\lambda e^{-\lambda t}$$

Thus, the exponential reliability function can be derived to be:


 * $$\begin{align} R(t)= & 1-\int_{0}^{t}\lambda {{e}^{-\lambda s}}ds \\

= & 1-[ 1-{{e}^{-\lambda \cdot t}}] \\ = & {{e}^{-\lambda \cdot t}} \\ \end{align}$$

The exponential failure rate function is:


 * $$\begin{align}

\lambda (t) =&  \frac{f(t)}{R(t)} \\ =& \frac{\lambda {e}^{-\lambda t}}{e^{-\lambda t}} \\ =& \lambda \end{align}$$

The exponential Mean-Time-To-Failure (MTTF) is given by:


 * $$\begin{align}

\mu = & \int_{0}^{\infty} t\cdot f(t)dt \\ = & \int_{0}^{\infty}{t \cdot {\lambda} \cdot e^{-\lambda t}}dt \\ = & \frac{1}{\lambda } \end{align}$$

This exact same methodology can be applied to any distribution given its $$pdf$$, with various degrees of difficulty depending on the complexity of $$f(t)$$.

Parameter Types
Distributions can have any number of parameters. Do note that as the number of parameters increases, so does the amount of data required for a proper fit. In general, most distributions used for reliability and life data analysis, the lifetime distributions, usually are limited to a maximum of three parameters. These three parameters are usually known as the scale parameter, the shape parameter and the location parameter.

Scale Parameter
The scale parameter is the most common type of parameter. All distributions in this reference have a scale parameter. In the case of one-parameter distributions, the sole parameter is the scale parameter. The scale parameter defines where the bulk of the distribution lies, or how stretched out the distribution is. In the case of the normal distribution, the scale parameter is the standard deviation.

Shape Parameter
The shape parameter, as the name implies, helps define the shape of a distribution. Some distributions, such as the exponential or normal, do not have a shape parameter since they have a predefined shape that does not change. In the case of the normal distribution, the shape is always the familiar bell shape. The effect of the shape parameter on a distribution is reflected in the shapes of the $$pdf$$, the reliability function and the failure rate function.

Location Parameter
The location parameter is used to shift a distribution in one direction or another. The location parameter, usually denoted as $$\gamma $$, defines the location of the origin of a distribution and can be either positive or negative. In terms of lifetime distributions, the location parameter represents a time shift.



This means that the inclusion of a location parameter for a distribution whose domain is normally $$[0,\infty]$$ will change the domain to $$[\gamma ,\infty]$$, where $$\gamma $$ can be either positive or negative. This can have some profound effects in terms of reliability. For a positive location parameter, this indicates that the reliability for that particular distribution is always 100% up to that point. In other words, a failure cannot occur before this time $$\gamma $$. Many engineers feel uncomfortable in saying that a failure will absolutely not happen before any given time. On the other hand, the argument can be made that almost all life distributions have a location parameter, although many of them may be negligibly small. Similarly, many people are uncomfortable with the concept of a negative location parameter, which states that failures theoretically occur before time zero. Realistically, the calculation of a negative location parameter is indicative of quiescent failures (failures that occur before a product is used for the first time) or of problems with the manufacturing, packaging or shipping processes. More attention will be given to the concept of the location parameter in subsequent discussions of the exponential and Weibull distributions, which are the lifetime distributions that most frequently employ the location parameter.

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:


 * $$f(t)=\frac{\beta}{\eta }( \frac{t-\gamma }{\eta } )^{\beta -1}{e}^{-(\tfrac{t-\gamma }{\eta }) ^{\beta}}$$

with three parameters $$\beta $$,  $$\eta $$  and  $$\gamma ,$$  where  $$\beta =$$  shape parameter,  $$\eta =$$  scale parameter and location parameter. If the location parameter, $$\gamma $$, is assumed to be zero, the distribution then becomes the two-parameter Weibull or:


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

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


 * $$f(t)=\frac{C}{\eta}(\frac{t}{\eta})^{C-1}e^{-(\frac{t}{\eta})^C}

$$

Chapter 6 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 ( $$\beta )$$ 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 Chapter 6.

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:


 * $$f(t)=\lambda {e}^{-\lambda(t-\gamma )}$$

(also known as the two-parameter exponential in this form) with two parameters, namely $$\lambda $$ and  $$\gamma .$$ If the location parameter, $$\gamma $$, is assumed to be zero, the distribution then becomes the one-parameter exponential or,


 * $$f(t)=\lambda {{e}^{-\lambda t}}$$

The exponential distribution and its characteristics are presented in more detail in Chapter 7.

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:


 * $$\begin{align}

f(t)= \frac{1}{\sigma \sqrt{2\pi }}{e^{-\tfrac{1}{2}(\tfrac{t-\mu }{\sigma })^2}} \end{align}$$

where,


 * $$\begin{align}

\mu = & \text{mean of the normal times to failure} \\ \sigma = & \text{standard deviation of the times to failure} \end{align}$$

The normal distribution and its characteristics are presented in more detail in Chapter 8.

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:


 * $$\begin{align}

& f(t)=\frac{1}{t{\sigma}_{T'}\sqrt{2\pi}}e^{-\tfrac{1}{2}(\tfrac{T'-{\mu'}}{\sigma_{T'}})^2}\\ & f(t)\ge 0,t>0,{{\sigma }_{T'}}>0 \\ & {T'}= \ln (t) \end{align} $$

Where,


 * $$\begin{align}

& {\mu'}= \text{mean of the natural logarithms of the times-to-failure} \\ & {\sigma_{T'}}= \text{standard deviation of the natural logarithms of the times to failure} \end{align}$$

The lognormal distribution and its characteristics are presented in more detail in Chapter 9.

Other Distributions
In addition to the distributions mentioned, the following additional distributions, even though not as frequently used in Life Data Analysis, have a variety of applications and can be found in many statistical references. They are included in Weibull++ as well as discussed in this reference.

The Mixed Weibull Distribution
The mixed Weibull distribution is commonly used for modeling the behavior of components or systems exhibiting multiple failure modes (mixed populations). It gives the global picture of the life of a product by mixing different Weibull distributions for different stages of the product’s life and is defined by:


 * $$f_{S}(t)=\sum_{i=1}^{S}p_{i}\frac{\beta_{i}}{\eta_{i}}(\frac{t}{\eta_{i}})^{\beta_{i}-1}e^{-(\frac{t}{\eta_{i}})^{\beta_{i}}}

$$

where the value of $$S$$  is equal to the number of subpopulations. Note that this results in a total of $$(3\cdot S-1)$$ parameters. In other words, each population has a portion or mixing weight for the $${{i}^{th}}$$ population, a $$\beta_{i}$$, or shape parameter for the $${{i}^{th}}$$ population and  or scale parameter $$\eta_{i}$$ for $${{i}^{th}}$$ population. Note that the parameters are reduced to $$(3\cdot S-1)$$, given the fact that the following condition can also be used:


 * $$\sum_{i=1}^{s}p_{i}=1$$

The mixed Weibull distribution and its characteristics are presented in more detail in Chapter 10.

The Generalized Gamma Distribution
While not as frequently used for modeling life data as the distributions discussed previously, the generalized gamma distribution does have the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters and also offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters μ, $$\sigma$$ and λ. The pdf of the distribution is given by,



f(x)=\begin{cases} \frac{|\lambda|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot {e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, &\text{if} \lambda \ne 0 \\

\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \lambda =0 \end{cases} $$

where Γ(x) is the gamma function, defined by:


 * $$\Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds$$

This distribution behaves as do other distributions based on the values of the parameters. For example, if λ = 1, the distribution is identical to the Weibull distribution. If both λ = 1 and σ = 1, the distribution is identical to the exponential distribution and for λ = 0, it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

The Gamma Distribution
The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, gamma distribution has applications in Bayesian analysis as a prior distribution and is also commonly used in queuing theory. The $$pdf$$ of the gamma distribution is given by:


 * $$\begin{align}

f(t)= & \frac{e^{kz-{e^{z}}}}{t\Gamma(k)} \\ z= & \ln{t}-\mu \end{align}$$

where:


 * $$\begin{align}

\mu = & \text{scale parameter} \\ k= & \text{shape parameter} \end{align}$$

where 0 $$0$$.

The gamma distribution and its characteristics are presented in more detail in Chapter 10.

The Logistic Distribution
The logistic distribution has a shape very similar to the normal distribution (i.e. bell shaped), but with heavier tails. Since the logistic distribution has closed form solutions for the reliability, $$cdf$$ and failure rate functions, it is sometimes preferred over the normal distribution, where these functions can only be obtained numerically. The $$pdf$$ of the logistic distribution is given by:


 * $$\begin{align}

f(t)= & \frac{e^z}{\sigma {(1+{e^z})^{2}}} \\ z= & \frac{t-\mu }{\sigma } \\ \sigma > & 0 \end{align}$$

where:


 * $$ \mu = \text{location parameter,also denoted as }$$ $$\overline{T}$$


 * $$ \sigma=\text{scale parameter} $$

The logistic distribution and its characteristics are presented in more detail in Chapter 10.

The Loglogistic Distribution
As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities. The $$pdf$$ of the loglogistic distribution is given by:


 * $$ \begin{align}

f(t)= & \frac{e^z}{\sigma t{(1+{e^z})^2}} \\ z= & \frac{T'-{\mu }'}{\sigma } \\ f(t)\ge & 0,t>0,{{\sigma }_{T'}}>0, \\ {T}'= & ln(t) \end{align}$$

where,


 * $$\begin{align}

\mu'= & \text{scale parameter} \\ \sigma_{T}=& \text{shape parameter} \end{align}$$

The loglogistic distribution and its characteristics are presented in more detail in Chapter 10.

The Gumbel Distribution
The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type 1) distribution. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (few weak units fail under low stress, while the rest fail at higher stresses). The Gumbel distribution could also be appropriate for modeling the life of products that experience very quick wear out after reaching a certain age. The $$pdf$$ of the Gumbel distribution is given by:


 * $$\begin{align}

f(t)= & \frac{1}{\sigma }{{e}^{z-{e^z}}} \\ z= &\frac{t-\mu }{\sigma } \\ f(T)\ge & 0,\sigma >0 \end{align}$$

where,


 * $$\begin{align}

\mu = & \text{location parameter} \\ \sigma = & \text{scale parameter} \end{align}$$

The Gumbel distribution and its characteristics are presented in more detail in Chapter 10.