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| ===The One-Parameter Exponential Distribution===
| | [[Category: For Deletion]] |
| The one-parameter exponential <math>pdf</math> is obtained by setting <math>\gamma =0</math>, and is given by:
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| ::<math> \begin{align}f(t)= & \lambda {{e}^{-\lambda t}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}t}},
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| & t\ge 0, \lambda >0,m>0
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| \end{align}
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| </math>
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| where:
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| ::<math>\lambda </math> = constant rate, in failures per unit of measurement, ''e.g'' failures per hour, per cycle, etc.,
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| ::<math>\lambda =\frac{1}{m}</math>,
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| ::<math>m</math> = mean time between failures, or to failure,
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| ::<math>t</math> = operating time, life, or age, in hours, cycles, miles, actuations, etc.
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| 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|
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| [19]]]:
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| :#The location parameter, <math>\gamma </math>, is zero.
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| :#The scale parameter is <math>\tfrac{1}{\lambda }=m</math>.
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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.
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| :#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>.
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| :#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.
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| :#As <math>t\to \infty </math> , <math>f(t)\to 0</math>.
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| :#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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