Template:Alta ld statistical prop func: Difference between revisions

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===Statistical Properties Summary===
#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Lognormal_Distribution]]
{{ald mean}}
 
{{ald sd}}
 
{{ald median}}
 
====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