Template:Lognormal distribution statistical properties - Revision history

Richard House: Redirected page to Lognormal Distribution Functions

2012-08-13T04:44:46Z

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	==Lognormal ~~Statistical Properties==~~		#REDIRECT [[Lognormal Distribution Functions]]
	~~{{lognormal distribution mean}}~~

	~~{{lognormal distribution median}}~~

	~~{{lognormal distribution mode}}~~

	~~{{lognormal distribution standard deviation}}~~

	~~{{lognormal distribution reliability function}}~~

	~~{{lognormal distribution conditional reliability}}~~

	~~{{lognormal distribution reliable life}}~~

	~~{{lognormal distribution failure rate function}}~~

2012-02-13T16:19:43Z

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	==Lognormal Statistical Properties==		==Lognormal Statistical Properties==
	{{ld mean}}		{{lognormal distribution mean}}

	{{ld median}}		{{lognormal distribution median}}

	{{ld mode}}		{{lognormal distribution mode}}

	{{ld standard deviation}}		{{lognormal distribution standard deviation}}

	{{ld lognormal reliability function}}		{{lognormal distribution reliability function}}

	{{ld lognormal conditional reliability}}		{{lognormal distribution conditional reliability}}

	{{ld lognormal reliable life}}		{{lognormal distribution reliable life}}

	{{ld lognormal failure rate function}}		{{lognormal distribution failure rate function}}

2012-02-10T23:22:45Z

← Older revision	Revision as of 23:22, 10 February 2012
(No difference)

2012-01-04T18:21:50Z

The Lognormal Failure Rate Function

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	{{ld lognormal reliable life}}		{{ld lognormal reliable life}}

	~~===The Lognormal Failure Rate Function===~~		{{ld lognormal failure rate function}}
	~~The~~ lognormal failure rate ~~is given by:~~


	::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{T\cdot {{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{T}'-{\mu }'}{{{\sigma }_{{{T}'}}}})}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}~~{{(\tfrac{t-{\mu~~ }~~'}{{{\sigma }_{{{T}'}}}})}^{2}}}}dt}</math>~~

	~~As with the reliability equations, standard normal tables will be required to solve for this function.~~

2012-01-04T18:21:16Z

The Lognormal Reliable Life

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	{{ld lognormal conditional reliability}}		{{ld lognormal conditional reliability}}

	~~===The Lognormal Reliable Life===~~		{{ld lognormal reliable life}}
	~~As there is no closed-form solution for the lognormal reliability equation, no closed-form solution exists for the~~ lognormal reliable life ~~either. In order to determine this value, one must solve the equation:~~


	~~::<math>{{R}_{T}}=\int_{\text{ln}(T)}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{s-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}~~}}~~ds</math>~~

	~~for <math>T</math> .~~

	===The Lognormal Failure Rate Function===		===The Lognormal Failure Rate Function===

2012-01-04T18:20:32Z

The Lognormal Conditional Reliability

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	{{ld lognormal reliability function}}		{{ld lognormal reliability function}}

	~~===The Lognormal Conditional Reliability===~~		{{ld lognormal conditional reliability}}
	~~The~~ lognormal conditional reliability ~~function is given by:~~

	::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{s-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{s-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}ds}~~</math>~~

	~~Once again, the use of standard normal tables is necessary to solve this equation, as no closed-form solution exists.~~

	===The Lognormal Reliable Life===		===The Lognormal Reliable Life===

2012-01-04T18:20:07Z

The Lognormal Reliability Function

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	{{ld standard deviation}}		{{ld standard deviation}}

	~~===The Lognormal Reliability Function===~~		{{ld lognormal reliability function}}
	~~The reliability for a mission of time <math>T</math> , starting at age 0, for the lognormal distribution is determined by:~~

	~~::<math>R(T)=\int_{T}^{\infty }f(t)dt</math>~~

	~~:or:~~

	~~::<math>R(T)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac~~{1}{~~{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>~~

	~~As with the normal distribution, 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. For interested readers, full explanations can be found in the references.

	===The Lognormal Conditional Reliability===		===The Lognormal Conditional Reliability===

2012-01-04T18:19:31Z

The Standard Deviation

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	{{ld mode}}		{{ld mode}}

	~~===The Standard Deviation===~~		{{ld standard deviation}}
	~~The standard deviation of the lognormal distribution, <math>{{\sigma }_{T}}</math> , is given by [18]:~~

	~~::<math>{~~{~~\sigma }_~~{~~T}}=\sqrt{\left( {{e}^{2{\mu }'+\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 Lognormal Reliability Function===		===The Lognormal Reliability Function===

2012-01-04T18:19:08Z

The Mode

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	{{ld median}}		{{ld median}}

	~~===The Mode===~~		{{ld mode}}
	~~The mode of the lognormal distribution, <math>\tilde~~{~~T}</math> , is given by [1]:~~

	~~::<math>\tilde~~{~~T}={{e}^{{\mu }'-\sigma _{{{T}'}}^{2}~~}}~~</math>~~

	===The Standard Deviation===		===The Standard Deviation===

2012-01-04T18:18:45Z

The Median

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	{{ld mean}}		{{ld mean}}

	~~===The Median===~~		{{ld median}}
	~~The median of the lognormal distribution, <math>\breve~~{~~T}</math> , is given by [18]:~~

	~~::<math>\breve~~{~~T}={{e}^{{{\mu }'}~~}}~~</math>~~

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