Weibull++ Standard Folio Data 1P-Exponential: Difference between revisions

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{{one parameter exp distribution}}
===The One-Parameter Exponential Distribution===
The one-parameter exponential <math>pdf</math> is obtained by setting <math>\gamma =0</math>, and is given by:
 
<math> \begin{align}f(t)= & \lambda {{e}^{-\lambda t}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}t}},
  & t\ge 0, \lambda >0,m>0
\end{align}
</math>
 
where:
 
This distribution requires the knowledge of only one parameter, <math>\lambda </math>, for its application. Some of the characteristics of the one-parameter exponential distribution are [[Appendix: Weibull References|
[19]]]:
#The location parameter, <math>\gamma </math>, is zero.
#The scale parameter is <math>\tfrac{1}{\lambda }=m</math>.
#As <math>\lambda </math> is decreased in value, the distribution is stretched out to the right, and as <math>\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>\lambda </math>.
#The distribution starts at <math>t=0</math> at the level of <math>f(t=0)=\lambda </math> and decreases thereafter exponentially and monotonically as <math>t</math> increases, and is convex.
#As <math>t\to \infty </math> , <math>f(t)\to 0</math>.
#The <math>pdf</math> can be thought of as a special case of the Weibull <math>pdf</math> with <math>\gamma =0</math>  and <math>\beta =1</math>.
 
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| valign="middle" | [http://www.reliawiki.com/index.php/The_Exponential_Distribution Exponential Distribution]
| valign="middle" | [http://www.reliawiki.com/index.php/The_Exponential_Distribution Exponential Distribution]

Revision as of 17:12, 8 February 2012

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Standard Folio Data 1P-Exponential
Weibull++

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].
Exponential Distribution
See Examples...



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