Template:Exponential Distribution Definition: Difference between revisions

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The exponential distribution is commonly used for components or systems exhibiting a ''constant failure rate''. Due to its simplicity, it has been widely employed, even in cases where it doesn't apply. In its most general case, the 2-parameter exponential distribution is defined by: <br>
The exponential distribution is commonly used for components or systems exhibiting a ''constant failure rate''. Due to its simplicity, it has been widely employed, even in cases where it doesn't apply. In its most general case, the 2-parameter exponential distribution is defined by:
 


::<math>\begin{align}
::<math>\begin{align}
f(t)=\lambda e^{-\lambda (t-\gamma)}  
f(t)=\lambda e^{-\lambda (t-\gamma)}  
\end{align}</math>
\end{align}\,\!</math>
 
<math> \lambda</math> is the constant failure rate in failures per unit of measurement (e.g., failures per hour, per cycle, etc.) and <math>\gamma</math> is the location parameter. In addition, <math>\lambda =\tfrac{1}{m}</math>, where <math>{m}</math> is the mean time between failures (or to failure).


If the location parameter, <span class="texhtml">γ</span>, is assumed to be zero, then the distribution becomes the 1-parameter exponential or: <br>
Where <math> \lambda\,\!</math> is the constant failure rate in failures per unit of measurement (e.g., failures per hour, per cycle, etc.) and <math>\gamma\,\!</math> is the location parameter. In addition, <math>\lambda =\tfrac{1}{m}\,\!</math>, where <math>{m}\,\!</math> is the mean time between failures (or to failure).


If the location parameter, <math>\gamma\,\!</math>, is assumed to be zero, then the distribution becomes the 1-parameter exponential or:


::<math>\begin{align}
::<math>\begin{align}
f(t)=\lambda e^{-\lambda t}  
f(t)=\lambda e^{-\lambda t}  
\end{align}</math>
\end{align}\,\!</math>


For a detailed discussion of this distribution, see [[The Exponential Distribution|The Exponential Distribution]].
For a detailed discussion of this distribution, see [[The Exponential Distribution|The Exponential Distribution]].

Latest revision as of 18:20, 24 September 2012

The exponential distribution is commonly used for components or systems exhibiting a constant failure rate. Due to its simplicity, it has been widely employed, even in cases where it doesn't apply. In its most general case, the 2-parameter exponential distribution is defined by:

[math]\displaystyle{ \begin{align} f(t)=\lambda e^{-\lambda (t-\gamma)} \end{align}\,\! }[/math]

Where [math]\displaystyle{ \lambda\,\! }[/math] is the constant failure rate in failures per unit of measurement (e.g., failures per hour, per cycle, etc.) and [math]\displaystyle{ \gamma\,\! }[/math] is the location parameter. In addition, [math]\displaystyle{ \lambda =\tfrac{1}{m}\,\! }[/math], where [math]\displaystyle{ {m}\,\! }[/math] is the mean time between failures (or to failure).

If the location parameter, [math]\displaystyle{ \gamma\,\! }[/math], is assumed to be zero, then the distribution becomes the 1-parameter exponential or:

[math]\displaystyle{ \begin{align} f(t)=\lambda e^{-\lambda t} \end{align}\,\! }[/math]

For a detailed discussion of this distribution, see The Exponential Distribution.