View source for Template:One parameter exp distribution

===The One-Parameter Exponential Distribution===
The one-parameter exponential <math>pdf</math> 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, ets.

This distribution requires the knowledge of only one parameter, <math>\lambda </math>, for its application. Some of the characteristics of the one-parameter exponential distribution are [[Appendix: Weibull 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 <math>pdf</math> can be thought of as a special case of the Weibull <math>pdf</math> with <math>\gamma =0</math>  and <math>\beta =1</math>.