Template:Erying-log: Difference between revisions

Latest revision as of 23:31, 16 August 2012

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-==Eyring-Lognormal==
+#REDIRECT [[Eyring_Relationship#Eyring-Lognormal]]
-<br>
-The  <math>pdf</math>  of the lognormal distribution is given by:
-<br>
-::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
-<br>
-where:
-<br>
-::<math>{T}'=\ln (T)</math>
-<br>
-::<math>T=\text{times-to-failure}</math>
-<br>
-and
-<br>
-•	 <math>\overline{{{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.
-<br>
-The Eyring-lognormal model can be obtained first by setting  <math>\breve{T}=L(V)</math>:
-<br>
-::<math>\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>
-<br>
-or:
-<br>
-<br>
-::<math>{{e}^{{{\overline{T}}^{\prime }}}}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>
-<br>
-Thus:
-<br>
-<br>
-::<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>
-<br>
-Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model  <math>pdf</math>:
-::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
-<br>
-{{eyring-log stat prop sum}}
-===Parameter Estimation===
-{{eyring-log mle}}