Life Distributions

We use the term Life Distributions to describe the collection of statististical probability distribution that we use in Reliability Engineering and Life Data Analysis.

Life 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 $$f(t)$$.

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 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.