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:
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]:
- The location parameter, [math]\displaystyle{ \gamma }[/math], is zero.
- The scale parameter is [math]\displaystyle{ \tfrac{1}{\lambda }=m }[/math].
- 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.
- 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].
- 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.
- As [math]\displaystyle{ t\to \infty }[/math] , [math]\displaystyle{ f(t)\to 0 }[/math].
- 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].
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