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 $$\beta =1\,\!$$. 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).

The 2-Parameter Exponential Distribution
The 2-parameter exponential pdf is given by:


 * $$f(t)=\lambda {{e}^{-\lambda (t-\gamma )}},f(t)\ge 0,\lambda >0,t\ge \gamma \,\!$$

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

The 1-Parameter Exponential Distribution
The 1-parameter exponential pdf is obtained by setting $$\gamma =0\,\!$$, and is given by:


 * $$ \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} \,\!$$

where:


 * $$\lambda \,\!$$ = constant rate, in failures per unit of measurement, (e.g., failures per hour, per cycle, etc.)


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

This distribution requires the knowledge of only one parameter, $$\lambda \,\!$$, for its application. Some of the characteristics of the 1-parameter exponential distribution are discussed in Kececioglu [19]:
 * The location parameter, $$\gamma \,\!$$, is zero.
 * The scale parameter is $$\tfrac{1}{\lambda }=m\,\!$$.
 * As $$\lambda \,\!$$ is decreased in value, the distribution is stretched out to the right, and as $$\lambda \,\!$$ 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, $$\lambda \,\!$$).
 * The distribution starts at $$t=0\,\!$$ at the level of $$f(t=0)=\lambda \,\!$$ and decreases thereafter exponentially and monotonically as $$t\,\!$$ increases, and is convex.
 * As $$t\to \infty \,\!$$, $$f(t)\to 0\,\!$$.
 * The pdf can be thought of as a special case of the Weibull pdf with $$\gamma =0\,\!$$ and $$\beta =1\,\!$$.