Template:Alta ld statistical prop func: Difference between revisions

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(Created page with '===Statistical Properties Summary=== ====The Mean or MTTF==== :• The mean of the lognormal distribution, <math>\bar{T}</math> , is given by: ::<math>\bar{T}={{e}^{\bar{{T}'}+\…')
 
 
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===Statistical Properties Summary===
#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Lognormal_Distribution]]
====The Mean or MTTF====
:• The mean of the lognormal distribution,  <math>\bar{T}</math> , is given by:
::<math>\bar{T}={{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</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 lognormal distribution,  <math>{{\sigma }_{T}}</math> , is given by:
::<math>{{\sigma }_{T}}=\sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}</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>
<br>
 
====The Median====
:• The median of the lognormal distribution is given by:
<br>
::<math>\breve{T}={{e}^{{{\bar{T}}^{\prime }}}}</math>
 
====The Mode====
:• The mode of the lognormal distribution is given by:
<br>
::<math>\tilde{T}={{e}^{{{\bar{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}}</math>
 
====Reliability Function====
For the lognormal distribution, the reliability for a mission of time  <math>T</math> , starting at age 0, is given by:
<br>
::<math>R(T)=\mathop{}_{T}^{\infty }f(t)dt</math>
<br>
:or:
<br>
::<math>R(T)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
<br>
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables.
<br>
<br>
====Lognormal Failure Rate====
The lognormal failure rate is given by:
<br>
::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{T}'{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}dt}</math>

Latest revision as of 03:08, 16 August 2012

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