Lognormal Statistical Properties - Revision history

Richard House: Redirected page to Lognormal Distribution Functions

2012-08-13T05:05:57Z

Redirected page to Lognormal Distribution Functions

← Older revision		Revision as of 05:05, 13 August 2012
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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}}~~

Harry Guo at 21:13, 13 February 2012

2012-02-13T21:13:15Z

← Older revision		Revision as of 21:13, 13 February 2012
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	==~~==The Mean or MTTF==~~==		==Lognormal Statistical Properties==
	~~The mean of the~~ lognormal distribution~~, <math>\mu </math> , is given by [18]:~~		{{lognormal distribution mean}}

	~~::<math>\mu =~~{{~~e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}~~}}~~</math>~~		{{lognormal distribution median}}

			{{lognormal distribution mode}}

	~~The mean of the natural logarithms of the times-to-failure, <math>\mu'</math> , in terms of <math>\bar~~{~~T}</math> and <math>{{\sigma }_~~{T}}~~</math> is givgen by:~~		{{lognormal distribution standard deviation}}

	~~::<math>~~{~~\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^~~{2}}}~~+1 \right)</math>~~		{{lognormal distribution reliability function}}

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

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

	~~====The Mode====~~		{{lognormal distribution failure rate function}}
	~~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 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 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.

Steve Sharp at 17:13, 29 June 2011

2011-06-29T17:13:09Z

Steve Sharp: Created page with '====The Mean or MTTF==== The mean of the lognormal distribution, $\mu$ , is given by [18]: $\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}$ The…'

2011-06-28T17:02:13Z

Created page with '====The Mean or MTTF==== The mean of the lognormal distribution, <math>\mu </math> , is given by [18]: <math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</math> The…'

New page

====The Mean or MTTF====
The mean of the lognormal distribution, <math>\mu </math> , is given by [18]:

<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</math>

The mean of the natural logarithms of the times-to-failure, <math>\mu'</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is givgen by:

<math>{\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>

====The 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 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 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 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.

← Older revision		Revision as of 17:13, 29 June 2011
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	The mean of the lognormal distribution, <math>\mu </math> , is given by [18]:		The mean of the lognormal distribution, <math>\mu </math> , is given by [18]:

	<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</math>		::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</math>


	The mean of the natural logarithms of the times-to-failure, <math>\mu'</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is givgen by:		The mean of the natural logarithms of the times-to-failure, <math>\mu'</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is givgen by:

	<math>{\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>		::<math>{\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>

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

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

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

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

	====The Standard Deviation====		====The Standard Deviation====
	The standard deviation of the lognormal distribution, <math>{{\sigma }_{T}}</math> , is given by [18]:		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>		::<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:		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>		::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>


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	The reliability for a mission of time <math>T</math> , starting at age 0, for the lognormal distribution is determined by:		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>		::<math>R(T)=\int_{T}^{\infty }f(t)dt</math>

	or:		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>		::<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.		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.

Lognormal Statistical Properties - Revision history

Richard House: Redirected page to Lognormal Distribution Functions

Harry Guo at 21:13, 13 February 2012

Steve Sharp at 17:13, 29 June 2011

Steve Sharp: Created page with '====The Mean or MTTF==== The mean of the lognormal distribution, \mu , is given by [18]: \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} The…'

Steve Sharp: Created page with '====The Mean or MTTF==== The mean of the lognormal distribution, $\mu$ , is given by [18]: $\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}$ The…'