Template:One parameter exp distribution

The One-Parameter Exponential Distribution
The one-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 one-parameter exponential distribution are [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 $$\beta =1$$.