View source for The Exponential Distribution

{{template:LDABOOK|7|The Exponential Distribution}}
The exponential distribution is a commonly used distribution in reliability engineering. Mathematically, it is a fairly simple distribution, which many times leads to its use in inappropriate situations. It is, in fact, a special case of the Weibull distribution where <math>\beta =1\,\!</math>. The exponential distribution is used to model the behavior of units that have a constant failure rate (or units that do not degrade with time or wear out).  

==Exponential Probability Density Function==
===The 2-Parameter Exponential Distribution===
The 2-parameter exponential ''pdf'' is given by:

::<math>f(t)=\lambda {{e}^{-\lambda (t-\gamma )}},f(t)\ge 0,\lambda >0,t\ge \gamma \,\!</math>

where <math>\gamma \,\!</math> is the location parameter.
Some of the characteristics of the 2-parameter exponential distribution are discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References|[19]]]:
*The location parameter, <math>\gamma \,\!</math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma \,\!</math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma \,\!</math> hours of operation, and cannot occur before.
*The scale parameter is <math>\tfrac{1}{\lambda }=\bar{t}-\gamma =m-\gamma \,\!</math>.
*The exponential ''pdf'' has no shape parameter, as it has only one shape.
*The distribution starts at <math>t=\gamma \,\!</math> at the level of <math>f(t=\gamma )=\lambda \,\!</math> and decreases thereafter exponentially and monotonically as <math>t\,\!</math> increases beyond <math>\gamma \,\!</math> and is convex.
*As <math>t\to \infty \,\!</math>, <math>f(t)\to 0\,\!</math>.

===The 1-Parameter Exponential Distribution===
The 1-parameter exponential ''pdf'' is obtained by setting <math>\gamma =0\,\!</math>, and is given by:

::<math> \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}
\,\!</math>

where:

::<math>\lambda \,\!</math> = constant rate, in failures per unit of measurement, (e.g., failures per hour, per cycle, etc.)

::<math>\lambda =\frac{1}{m}\,\!</math>
::<math>m\,\!</math> = mean time between failures, or to failure
::<math>t\,\!</math> = operating time, life, or age, in hours, cycles, miles, actuations, etc.

This distribution requires the knowledge of only one parameter, <math>\lambda \,\!</math>, for its application. Some of the characteristics of the 1-parameter exponential distribution are discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References| [19]]]:
*The location parameter, <math>\gamma \,\!</math>, is zero.
*The scale parameter is <math>\tfrac{1}{\lambda }=m\,\!</math>.
*As <math>\lambda \,\!</math> is decreased in value, the distribution is stretched out to the right, and as <math>\lambda \,\!</math> is increased, the distribution is pushed toward the origin.
*This distribution has no shape parameter as it has only one shape, (i.e., the exponential, and the only parameter it has is the failure rate, <math>\lambda \,\!</math>).
*The distribution starts at <math>t=0\,\!</math> at the level of <math>f(t=0)=\lambda \,\!</math> and decreases thereafter exponentially and monotonically as <math>t\,\!</math> increases, and is convex.
*As <math>t\to \infty \,\!</math>, <math>f(t)\to 0\,\!</math>.
*The ''pdf'' can be thought of as a special case of the Weibull ''pdf'' with <math>\gamma =0\,\!</math> and <math>\beta =1\,\!</math>.

==Exponential Distribution Functions==
{{:Exponential Distribution Functions}}

==Characteristics of the Exponential Distribution==
{{:Exponential Distribution Characteristics}}





==Exponential Distribution Examples==
{{:Exponential Distribution Examples}}