Template:One parameter exp distribution: Difference between revisions
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The one-parameter exponential <math>pdf</math> is obtained by setting <math>\gamma =0</math>, and is given by: | The one-parameter exponential <math>pdf</math> is obtained by setting <math>\gamma =0</math>, and is given by: | ||
::<math> \begin{align}f( | ::<math> \begin{align}f(t)= & \lambda {{e}^{-\lambda t}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}t}}, | ||
& T\ge 0, \lambda >0,m>0 | & T\ge 0, \lambda >0,m>0 | ||
\end{align} | \end{align} | ||
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:#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. | :#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>. | :#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> | :#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> | :#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>. | :#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>. | ||
Revision as of 23:09, 7 February 2012
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].