Template:Alta ld statistical prop func: Difference between revisions

Latest revision as of 03:08, 16 August 2012

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-===Statistical Properties Summary===
+#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Lognormal_Distribution]]
-{{ald mean}}
-{{ald sd}}
-====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>