Template:Two parameter exp distribution: Difference between revisions
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The two-parameter exponential ''pdf'' is given by: | The two-parameter exponential ''pdf'' is given by: | ||
::<math>f( | ::<math>f(t)=\lambda {{e}^{-\lambda (t-\gamma )}},f(t)\ge 0,\lambda >0,t\ge 0\text{ or }\gamma </math> | ||
where <math>\gamma </math> is the location parameter. | where <math>\gamma </math> is the location parameter. | ||
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[19]]]: | [19]]]: | ||
#The location parameter, <math>\gamma </math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma </math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma </math> hours of operation, and cannot occur before. | #The location parameter, <math>\gamma </math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma </math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma </math> hours of operation, and cannot occur before. | ||
#The scale parameter is <math>\tfrac{1}{\lambda }=\bar{ | #The scale parameter is <math>\tfrac{1}{\lambda }=\bar{t}-\gamma =m-\gamma </math>. | ||
#The exponential <math>pdf</math> has no shape parameter, as it has only one shape. | #The exponential <math>pdf</math> has no shape parameter, as it has only one shape. | ||
#The distribution starts at <math> | #The distribution starts at <math>t=\gamma </math> at the level of <math>f(t=\gamma )=\lambda </math> and decreases thereafter exponentially and monotonically as <math>t</math> increases beyond <math>\gamma </math> and is convex. | ||
#As <math> | #As <math>t\to \infty </math>, <math>f(t)\to 0</math>. | ||
<br> | <br> | ||
Revision as of 23:08, 7 February 2012
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]:
- 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.
- The scale parameter is [math]\displaystyle{ \tfrac{1}{\lambda }=\bar{t}-\gamma =m-\gamma }[/math].
- The exponential [math]\displaystyle{ pdf }[/math] has no shape parameter, as it has only one shape.
- 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.
- As [math]\displaystyle{ t\to \infty }[/math], [math]\displaystyle{ f(t)\to 0 }[/math].