Template:Exponential Distribution Definition: Difference between revisions
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::<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> | ||
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). | 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, < | 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.