Template:Two parameter exp distribution

The Two-Parameter Exponential Distribution

The two-parameter exponential pdf is given by:

[math]\displaystyle{ f(t)=\lambda {{e}^{-\lambda (t-\gamma )}},f(t)\ge 0,\lambda \gt 0,t\ge 0\text{ or }\gamma }[/math]

where [math]\displaystyle{ \gamma }[/math] is the location parameter. Some of the characteristics of the two-parameter exponential distribution are [19]:

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