Template:Exponential Probability Density Function

Exponential Probability Density Function

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].

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]:

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].