ALTA ALTA Standard Folio Data Eyring-Exponential: Difference between revisions

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The <math>pdf</math> of the 1-parameter exponential distribution is given by:
The <math>pdf</math> of the 1-parameter exponential distribution is given by:
<br>
<br>
<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}}</math>
<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}}</math>
<br>
<br>
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
<br>
<br>
<math>\lambda =\frac{1}{m}</math>
<math>\lambda =\frac{1}{m}</math>
<br>
<br>
thus:
thus:
<br>
<br>
<math>f(t)=\frac{1}{m}\cdot {{e}^{-\tfrac{t}{m}}}</math>
<math>f(t)=\frac{1}{m}\cdot {{e}^{-\tfrac{t}{m}}}</math>
<br>
<br>
The Eyring-exponential model  <math>pdf</math>  can then be obtained by setting  <math>m=L(V)</math>  in Eqn. (eyring):  
The Eyring-exponential model  <math>pdf</math>  can then be obtained by setting  <math>m=L(V)</math>  in Eqn. (eyring):  
<br>
<br>
<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
<br>
<br>
and substituting for  <math>m</math>  in Eqn. (pdfexpm2):
and substituting for  <math>m</math>  in Eqn. (pdfexpm2):
<br>
<br>
<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
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Revision as of 21:52, 10 February 2012

Webnotes-alta.png

Reliability Web Notes

Standard Folio Data Eyring-Exponential
ALTA

The [math]\displaystyle{ pdf }[/math] of the 1-parameter exponential distribution is given by:
[math]\displaystyle{ f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}} }[/math]
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
[math]\displaystyle{ \lambda =\frac{1}{m} }[/math]
thus:
[math]\displaystyle{ f(t)=\frac{1}{m}\cdot {{e}^{-\tfrac{t}{m}}} }[/math]
The Eyring-exponential model [math]\displaystyle{ pdf }[/math] can then be obtained by setting [math]\displaystyle{ m=L(V) }[/math] in Eqn. (eyring):
[math]\displaystyle{ m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}} }[/math]
and substituting for [math]\displaystyle{ m }[/math] in Eqn. (pdfexpm2):
[math]\displaystyle{ f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}} }[/math]

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