Template:Alta al - Revision history

Richard House: Redirected page to Arrhenius Relationship#Arrhenius-Lognormal

2012-08-16T05:28:34Z

Redirected page to Arrhenius Relationship#Arrhenius-Lognormal

← Older revision		Revision as of 05:28, 16 August 2012
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	==Arrhenius-Lognormal==		#REDIRECT [[Arrhenius_Relationship#Arrhenius-Lognormal]]
	~~<br>~~
	~~The <math>pdf</math> of the lognormal distribution is given by:~~
	~~<br>~~
	~~<br>~~
	~~::<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>~~

	~~<br>~~
	~~where:~~

	~~<br>~~
	~~::<math>{T}'=\ln(T) </math>~~

	~~<br>~~
	~~and:~~
	~~<br>~~
	~~• <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.~~
	~~<br>~~
	~~<br>~~
	~~The median of the lognormal distribution is given by:~~

	~~<br>~~
	~~::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>~~

	~~<br>~~

	~~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>~~

	~~<br>~~
	Note that in the Arrhenius-lognormal <math>pdf</math>, 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).
	~~<br>~~
	~~{{alta al stat prop sum}}~~

	~~===Parameter Estimation===~~
	~~<br>~~
	~~{{arrhenius-log mle}}~~

Athanasios Gerokostopoulos: /* Parameter Estimation */

2012-02-14T00:46:16Z

Parameter Estimation

← Older revision		Revision as of 00:46, 14 February 2012
Line 52:		Line 52:

	===Parameter Estimation===		===Parameter Estimation===
	~~<br>~~
	<br>		<br>
	{{arrhenius-log mle}}		{{arrhenius-log mle}}

Athanasios Gerokostopoulos at 00:11, 14 February 2012

2012-02-14T00:11:27Z

← Older revision		Revision as of 00:11, 14 February 2012
Line 47:		Line 47:

	<br>		<br>
	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).		Note that in the Arrhenius-lognormal <math>pdf</math>, 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).
	<br>		<br>
	{{alta al stat prop sum}}		{{alta al stat prop sum}}

Athanasios Gerokostopoulos: /* Arrhenius-Lognormal */

2012-02-14T00:10:22Z

Arrhenius-Lognormal

← Older revision		Revision as of 00:10, 14 February 2012
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	<br>		<br>

	The Arrhenius-lognormal model <math>pdf</math> can be obtained first by setting <math>\breve{T}=L(V)</math> ~~in Eqn. (arrhenius)~~. Therefore:		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>		::<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
Line 42:		Line 42:


	Substituting ~~Eqn. (arrh-logn-mean)~~ into ~~Eqn. (arrh-logn-~~pdf) yields the Arrhenius-lognormal model <math>pdf</math> or:		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>		::<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>

Athanasios Gerokostopoulos: /* Arrhenius-Lognormal */

2012-02-14T00:08:37Z

Arrhenius-Lognormal

← Older revision		Revision as of 00:08, 14 February 2012
Line 18:		Line 18:

	• <math>{T}'=</math> mean of the natural logarithms of the times-to-failure.		• <math>{T}'=</math> mean of the natural logarithms of the times-to-failure.

	~~• <math>T=</math> times-to-failure.~~

	• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.		• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.

Nicolette Young: /* Maximum Likelihood Estimation Method */

2012-01-12T22:13:19Z

Maximum Likelihood Estimation Method

@@ Line 56: / Line 56: @@
 <br>
 <br>
-====Maximum Likelihood Estimation Method====
+{{arrhenius-log mle}}

Nicolette Young: /* Arrhenius-Lognormal Statistical Properties Summary */

2012-01-12T21:49:11Z

Arrhenius-Lognormal Statistical Properties Summary

@@ Line 51: / Line 51: @@
 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).
 <br>
-===Arrhenius-Lognormal Statistical Properties Summary===
+{{alta al stat prop sum}}
 ===Parameter Estimation===
 <br>

Nicolette Young: Created page with '==Arrhenius-Lognormal==
The $pdf$ of the lognormal distribution is given by:

:: $f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\…'$

2012-01-12T21:48:15Z

Created page with '==Arrhenius-Lognormal== The <math>pdf</math> of the lognormal distribution is given by: ::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\…'

New page

==Arrhenius-Lognormal==
 
The <math>pdf</math> of the lognormal distribution is given by:
 
 
::<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>

 
where:

 
::<math>{T}'=\ln(T) </math>

 
and:
 
• <math>T=</math> times-to-failure.

• <math>{T}'=</math> mean of the natural logarithms of the times-to-failure.

• <math>T=</math> times-to-failure.

• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.
 
 
The median of the lognormal distribution is given by:

 
::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>

 

The Arrhenius-lognormal model <math>pdf</math> can be obtained first by setting <math>\breve{T}=L(V)</math> in Eqn. (arrhenius). 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 Eqn. (arrh-logn-mean) into Eqn. (arrh-logn-pdf) 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>

 
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).
 
===Arrhenius-Lognormal Statistical Properties Summary===
====The Mean====
 
• The mean life of the Arrhenius-lognormal model (mean of the times-to-failure), <math>\bar{T}</math> , is given by:

 
::<math>\begin{align}
& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{\ln (C)+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
\end{align}</math>

 
• The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:

 
::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>

====The Standard Deviation====
 
• The standard deviation of the Arrhenius-lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}</math> , is given by:

 
::<math>\begin{align}
& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\
& = & \sqrt{\left( {{e}^{2\left( \ln (C)+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
\end{align}</math>

 
• The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:
 
::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>
 
====The Mode====
 
• The mode of the Arrhenius-lognormal model is given by:

::<math>\begin{align}
& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{\ln (C)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
\end{align}</math>

====Arrhenius-Lognormal Reliability Function====
 
The reliability for a mission of time <math>T</math> , starting at age 0, for the Arrhenius-lognormal model is determined by:
 
::<math>R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt</math>

vor:
 
::<math>R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
 
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability, we will not discuss manual solution methods.
 
====Reliable Life====
 
For the Arrhenius-lognormal model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is estimated by first solving the reliability equation with respect to time, as follows,
..
 
where:

 
::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right]</math>

 
and:

 
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

 
Since <math>{T}'=\ln (T)</math> the reliable life, <math>{{t}_{R}},</math> is given by:

 
::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}</math>

====Arrhenius-Lognormal Failure Rate====
 
The Arrhenius-lognormal failure rate is given by:
 
::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>

====When Using the Lognormal Distribution in ALTA====
 
The parameters returned for the Arrhenius-lognormal distribution are always <math>{{\sigma }_{{{T}'}}},</math> <math>C,</math> and <math>B.</math> The returned <math>{{\sigma }_{{{T}'}}}</math> is always the square root of the variance of the natural logarithms to failure. Also, if the "Show Scale Parameter" option is checked (on the Data Sheet tab in the User Setup), the returned mean value is always the mean of the natural logarithms of the times-to-failure, given by Eqn. (arrh-logn-mean). Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as parameters of the distribution, and are thus the mean (a function of stress as it can be seen in Eqn. (arrh-logn-mean)) and standard deviation of the natural logarithms of the data. The mean life value of the times-to-failure, as well as the standard deviation of times-to-failure (not the parameter) can be obtained through the Function Wizard in ALTA.
 
 
===Parameter Estimation===
 
 
====Maximum Likelihood Estimation Method====
 
The lognormal log-likelihood function for the Arrhenius-lognormal model is as follows:

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\
& & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\
& & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}</math>

 
where:

 
::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>

 
::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>

 
and:
 
• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.

• <math>{{N}_{i}}</math> is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.

• .. is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

• <math>B</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

• <math>C</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).

• <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.

• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.

• <math>S</math> is the number of groups of suspension data points.

• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.

• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.

• <math>FI</math> is the number of interval data groups.

• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.

• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.

• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.

The solution (parameter estimates) will be found by solving for <math>{{\widehat{\sigma }}_{{{T}'}}},</math> <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:
 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\frac{1}{{{V}_{i}}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) \\
& & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}</math>

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C\cdot \sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) \\
& & +\frac{1}{C\cdot {{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }C(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
& & \\
& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-\ln (C)-\tfrac{B}{{{V}_{i}}} \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
& & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}</math>

 
and:

 
::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>

 
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

Template:Alta al - Revision history

Richard House: Redirected page to Arrhenius Relationship#Arrhenius-Lognormal

Athanasios Gerokostopoulos: /* Parameter Estimation */

Athanasios Gerokostopoulos at 00:11, 14 February 2012

Athanasios Gerokostopoulos: /* Arrhenius-Lognormal */

Athanasios Gerokostopoulos: /* Arrhenius-Lognormal */

Nicolette Young: /* Maximum Likelihood Estimation Method */

Nicolette Young: /* Arrhenius-Lognormal Statistical Properties Summary */

Nicolette Young: Created page with '==Arrhenius-Lognormal== The pdf of the lognormal distribution is given by: ::f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\…'

Nicolette Young: Created page with '==Arrhenius-Lognormal==
The $pdf$ of the lognormal distribution is given by:

:: $f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\…'$