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==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:
::<math>f(t)=\lambda e^{-\lambda t}</math>
In this definition, note that <math>t</math> is our random variable which represents time and the Greek letter <math>\lambda </math> (lambda) represents what is commonly referred to as the ''parameter'' of the distribution. Depending on the value of <math>\lambda ,</math>  <math>f(t)</math> 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:
<br>
::<math>f(t)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}(\frac{t-\mu}{\sigma})^2}</math>
<br>
<math>\mu</math>, the mean, and <math>\sigma ,</math> 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 <math>f(t)</math> is fully defined and we can obtain any value for <math>f(t)</math> given any value of <math>t</math>.
<br>
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 <math>t</math> after the value of the distribution parameter or parameters have been estimated from data.<sup></sup>
For example, we know that the exponential distribution <math>pdf</math> is given by:
<br>
::<math>f(t)=\lambda e^{-\lambda t}</math>
<br>
Thus, the exponential reliability function can be derived to be:
<br>
::<math>\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}</math>
The exponential failure rate function is:
::<math>\begin{align}
  \lambda (t)  =&  \frac{f(t)}{R(t)} \\
  =&  \frac{\lambda {e}^{-\lambda t}}{e^{-\lambda t}} \\
  =& \lambda 
\end{align}</math>
The exponential Mean-Time-To-Failure (MTTF) is given by:
::<math>\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}</math>
This exact same methodology can be applied to any distribution given its <math>pdf</math>, with various degrees of difficulty depending on the complexity of <math>f(t)</math>.

Revision as of 23:03, 3 January 2012

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