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| ==Exponential Probability Density Function==
| | #REDIRECT [[The Exponential Distribution]] |
| ===The Two-Parameter Exponential Distribution===
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| The two-parameter exponential ''pdf'' is given by:
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| ::<math>f(T)=\lambda {{e}^{-\lambda (T-\gamma )}},f(T)\ge 0,\lambda >0,T\ge 0\text{ or }\gamma </math>
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| where <math>\gamma </math> is the location parameter.
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| Some of the characteristics of the two-parameter exponential distribution are [19]:
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| #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.
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| #The scale parameter is <math>\tfrac{1}{\lambda }=\bar{T}-\gamma =m-\gamma </math>.
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| #The exponential <math>pdf</math> has no shape parameter, as it has only one shape.
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| #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.
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| #As <math>T\to \infty </math>, <math>f(T)\to 0</math>.
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| <br>
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| ===The One-Parameter Exponential Distribution===
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| 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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| 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 [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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