Template:Alta ld statistical prop func - Revision history

Richard House: Redirected page to Distributions Used in Accelerated Testing#The Lognormal Distribution

2012-08-16T03:08:30Z

Redirected page to Distributions Used in Accelerated Testing#The Lognormal Distribution

← Older revision		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}}~~

	~~{{ald median}}~~

	~~{{ald mode}}~~

	~~{{ald rf}}~~

	~~{{ald failure rate}}~~

Nicolette Young: /* Lognormal Failure Rate */

2012-01-12T18:51:47Z

Lognormal Failure Rate

← Older revision		Revision as of 18:51, 12 January 2012
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	{{ald rf}}		{{ald rf}}

	~~====Lognormal Failure Rate====~~		{{ald 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>~~

Nicolette Young: /* Reliability Function */

2012-01-12T18:51:28Z

Reliability Function

← Older revision		Revision as of 18:51, 12 January 2012
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	{{ald mode}}		{{ald mode}}

	~~====Reliability Function====~~		{{ald rf}}
	~~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====		====Lognormal Failure Rate====
	The lognormal failure rate is given by:		The lognormal failure rate is given by:
	<br>		<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>		::<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>

Nicolette Young: /* The Mode */

2012-01-12T18:51:08Z

The Mode

← Older revision		Revision as of 18:51, 12 January 2012
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	{{ald median}}		{{ald median}}

	~~====The Mode====~~		{{ald mode}}
	~~:• The mode of the lognormal distribution is given by:~~
	~~<br>~~
	~~::<math>\tilde~~{~~T}=~~{~~{e}^{{{\bar{T}}^{\prime }}-\sigma _{{{T}'}}^{2}~~}}~~</math>~~

	====Reliability Function====		====Reliability Function====

Nicolette Young: /* The Median */

2012-01-12T18:50:52Z

The Median

← Older revision		Revision as of 18:50, 12 January 2012
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	{{ald sd}}		{{ald sd}}

	~~====The Median====~~		{{ald median}}
	~~:• The median of the lognormal distribution is given by:~~
	~~<br>~~
	~~::<math>\breve{T}=~~{{~~e}^{{{\bar{T}}^{\prime }}~~}}~~</math>~~

	====The Mode====		====The Mode====

Nicolette Young: /* The Standard Deviation */

2012-01-12T18:50:37Z

The Standard Deviation

← Older revision		Revision as of 18:50, 12 January 2012
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	{{ald mean}}		{{ald mean}}

	~~====The Standard Deviation====~~		{{ald sd}}
	~~:• 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====

Nicolette Young: /* The Mean or MTTF */

2012-01-12T18:49:58Z

The Mean or MTTF

← Older revision		Revision as of 18:49, 12 January 2012
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	===Statistical Properties Summary===		===Statistical Properties Summary===
	~~====The Mean or MTTF====~~		{{ald mean}}
	~~:• 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====

Nicolette Young: Created page with '===Statistical Properties Summary=== ====The Mean or MTTF==== :• The mean of the lognormal distribution, $\bar{T}$ , is given by: :: $\bar{T}={{e}^{\bar{{T}'}+\…'$

2012-01-12T18:49:39Z

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}'}+\…'

New page

===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}'}+\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>
 

====The Median====
:• The median of the lognormal distribution is given by:
 
::<math>\breve{T}={{e}^{{{\bar{T}}^{\prime }}}}</math>

====The Mode====
:• The mode of the lognormal distribution is given by:
 
::<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:
 
::<math>R(T)=\mathop{}_{T}^{\infty }f(t)dt</math>
 
:or:
 
::<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>
 
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables.
 
 
====Lognormal Failure Rate====
The lognormal failure rate is given by:
 
::<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>