Template:One parameter exp distribution

The One-Parameter Exponential Distribution

The one-parameter exponential [math]\displaystyle{ pdf }[/math] is obtained by setting [math]\displaystyle{ \gamma =0 }[/math], and is given by:

[math]\displaystyle{ \begin{align}f(t)= & \lambda {{e}^{-\lambda t}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}t}}, & t\ge 0, \lambda \gt 0,m\gt 0 \end{align} }[/math]

where:

[math]\displaystyle{ \lambda }[/math] = constant rate, in failures per unit of measurement, e.g failures per hour, per cycle, etc.,

[math]\displaystyle{ \lambda =\frac{1}{m} }[/math],

[math]\displaystyle{ m }[/math] = mean time between failures, or to failure,

[math]\displaystyle{ t }[/math] = operating time, life, or age, in hours, cycles, miles, actuations, etc.

This distribution requires the knowledge of only one parameter, [math]\displaystyle{ \lambda }[/math], for its application. Some of the characteristics of the one-parameter exponential distribution are [19]:

1. The location parameter, [math]\displaystyle{ \gamma }[/math], is zero.
2. The scale parameter is [math]\displaystyle{ \tfrac{1}{\lambda }=m }[/math].
3. As [math]\displaystyle{ \lambda }[/math] is decreased in value, the distribution is stretched out to the right, and as [math]\displaystyle{ \lambda }[/math] is increased, the distribution is pushed toward the origin.
4. 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]\displaystyle{ \lambda }[/math].
5. The distribution starts at [math]\displaystyle{ t=0 }[/math] at the level of [math]\displaystyle{ f(t=0)=\lambda }[/math] and decreases thereafter exponentially and monotonically as [math]\displaystyle{ t }[/math] increases, and is convex.
6. As [math]\displaystyle{ t\to \infty }[/math] , [math]\displaystyle{ f(t)\to 0 }[/math].
7. The [math]\displaystyle{ pdf }[/math] can be thought of as a special case of the Weibull [math]\displaystyle{ pdf }[/math] with [math]\displaystyle{ \gamma =0 }[/math] and [math]\displaystyle{ \beta =1 }[/math].