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==Arrhenius-Lognormal==
#REDIRECT [[Arrhenius_Relationship#Arrhenius-Lognormal]]
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The  <math>pdf</math>  of the lognormal distribution is given by:
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::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
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where:
 
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::<math>{T}'=\ln(T) </math>
 
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and:
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• <math>T=</math>  times-to-failure.
 
• <math>{T}'=</math>  mean of the natural logarithms of the times-to-failure.
 
• <math>{{\sigma }_{{{T}'}}}=</math>  standard deviation of the natural logarithms of the times-to-failure.
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The median of the lognormal distribution is given by:
 
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::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>
 
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The Arrhenius-lognormal model  <math>pdf</math>  can be obtained first by setting <math>\breve{T}=L(V)</math>. Therefore:
 
::<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
 
or:
 
::<math>{{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}}</math>
 
Thus:
 
::<math>{{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V}</math>
 
 
Substituting the above equation into the lognormal <math>pdf</math> yields the Arrhenius-lognormal model  <math>pdf</math>  or:
 
::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
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Note that in Eqn. (arrh-logn-pdf), it was assumed that the standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}},</math>  is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( <math>{{\sigma }_{{{T}'}}}</math>  is the shape parameter of the lognormal distribution).
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{{alta al stat prop sum}}
 
===Parameter Estimation===
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{{arrhenius-log mle}}

Latest revision as of 05:28, 16 August 2012

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