Lognormal Distribution Functions - Revision history

Lisa Hacker at 21:43, 18 September 2023

2023-09-18T21:43:39Z

← Older revision		Revision as of 21:43, 18 September 2023
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	<noinclude>{{Navigation box}}[[Category:Shared Articles]]		<noinclude>{{Navigation box}}[[Category:Shared Articles]]
	''This article also appears in the [~~[The_Lognormal_Distribution\|~~Life ~~Data Analysis Reference]~~] and [~~[Distributions_Used_in_Accelerated_Testing\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] ~~books~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference] and [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference].'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean of the lognormal distribution, <math>\mu \,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:		The mean of the lognormal distribution, <math>\mu \,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

Harry Guo: /* The Standard Deviation */

2013-08-21T23:28:24Z

The Standard Deviation

← Older revision		Revision as of 23:28, 21 August 2013
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	The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:		The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

	::<math>{\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)-\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\!</math>		::<math>{\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\!</math>

	The standard deviation of the natural logarithms of the times-to-failure, <math>{\sigma}'\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{\sigma}\,\!</math> is given by:		The standard deviation of the natural logarithms of the times-to-failure, <math>{\sigma}'\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{\sigma}\,\!</math> is given by:

Richard House at 18:10, 24 September 2012

2012-09-24T18:10:55Z

Richard House at 16:05, 5 September 2012

2012-09-05T16:05:54Z

← Older revision		Revision as of 16:05, 5 September 2012
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	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean of the lognormal distribution, <math>\mu \,\!</math> , is discussed in [[Appendix:_Life_Data_Analysis_References\|[18]]]:		The mean of the lognormal distribution, <math>\mu \,\!</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

	::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>		::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>
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	===The Median===		===The Median===
	The median of the lognormal distribution, <math>\breve{T}</math> , is discussed in [[Appendix:_Life_Data_Analysis_References\|[18]]]:		The median of the lognormal distribution, <math>\breve{T}</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

	::<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 discussed in [[Appendix:_Life_Data_Analysis_References\|[1]]]:		The mode of the lognormal distribution, <math>\tilde{T}</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

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

	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math> , is discussed in [[Appendix:_Life_Data_Analysis_References\|[18]]]:		The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

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

Richard House at 20:43, 4 September 2012

2012-09-04T20:43:29Z

← Older revision		Revision as of 20:43, 4 September 2012
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	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean of the lognormal distribution, <math>\mu \,\!</math> , is ~~given by~~ [[Appendix:_Life_Data_Analysis_References\|[18]]]:		The mean of the lognormal distribution, <math>\mu \,\!</math> , is discussed in [[Appendix:_Life_Data_Analysis_References\|[18]]]:

	::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>		::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>
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	===The Median===		===The Median===
	The median of the lognormal distribution, <math>\breve{T}</math> , is ~~given by~~ [[Appendix:_Life_Data_Analysis_References\|[18]]]:		The median of the lognormal distribution, <math>\breve{T}</math> , is discussed in [[Appendix:_Life_Data_Analysis_References\|[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~~ [[Appendix:_Life_Data_Analysis_References\|[1]]]:		The mode of the lognormal distribution, <math>\tilde{T}</math> , is discussed in [[Appendix:_Life_Data_Analysis_References\|[1]]]:

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

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

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

Richard House at 20:42, 4 September 2012

2012-09-04T20:42:08Z

← Older revision		Revision as of 20:42, 4 September 2012
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	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean of the lognormal distribution, <math>\mu \,\!</math> , is given by [[Appendix: ~~Weibull References~~\|[18]]]:		The mean of the lognormal distribution, <math>\mu \,\!</math> , is given by [[Appendix:_Life_Data_Analysis_References\|[18]]]:

	::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>		::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>
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	===The Median===		===The Median===
	The median of the lognormal distribution, <math>\breve{T}</math> , is given by [[Appendix: ~~Weibull References~~\|[18]]]:		The median of the lognormal distribution, <math>\breve{T}</math> , is given by [[Appendix:_Life_Data_Analysis_References\|[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 [[Appendix: ~~Weibull References~~\|[1]]]:		The mode of the lognormal distribution, <math>\tilde{T}</math> , is given by [[Appendix:_Life_Data_Analysis_References\|[1]]]:

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

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

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

Richard House: /* The Lognormal Reliability Function */

2012-08-22T03:03:38Z

The Lognormal Reliability Function

← Older revision		Revision as of 03:03, 22 August 2012
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	===The Lognormal Reliability Function===		===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:		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(x)dx</math>		::<math>R(t)=\int_{t}^{\infty }f(x)dx</math>

Richard House: /* The Standard Deviation */

2012-08-22T03:03:19Z

The Standard Deviation

← Older revision		Revision as of 03:03, 22 August 2012
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	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation of the lognormal distribution, <math>{\sigma }_{T}</math> , is given by [[Appendix: Weibull References\|[18]]]:		The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math> , is given by [[Appendix: Weibull References\|[18]]]:

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

	The standard deviation of the natural logarithms of the times-to-failure, <math>{\sigma}'</math> , in terms of <math>\bar{T}</math> and <math>{\sigma}</math> is given by:		The standard deviation of the natural logarithms of the times-to-failure, <math>{\sigma}'\,\!</math> , in terms of <math>\bar{T}\,\!</math> and <math>{\sigma}\,\!</math> is given by:

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

Richard House: /* The Mean or MTTF */

2012-08-22T03:02:43Z

The Mean or MTTF

← Older revision		Revision as of 03:02, 22 August 2012
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	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean of the lognormal distribution, <math>\mu </math> , is given by [[Appendix: Weibull References\|[18]]]:		The mean of the lognormal distribution, <math>\mu \,\!</math> , is given by [[Appendix: Weibull References\|[18]]]:

	::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>		::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{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}}</math> is given 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}}\,\!</math> is given by:

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

Kate Racaza: added category

2012-08-15T05:04:36Z

added category

← Older revision		Revision as of 05:04, 15 August 2012
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	<noinclude>{{Navigation box}}		<noinclude>{{Navigation box}}[[Category:Shared Articles]]
	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===

← Older revision		Revision as of 18:10, 24 September 2012
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	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean of the lognormal distribution, <math>\mu \,\!</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:		The mean of the lognormal distribution, <math>\mu \,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

	::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}</math>		::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{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}}\,\!</math> is given 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}}\,\!</math> is given by:

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

	===The Median===		===The Median===
	The median of the lognormal distribution, <math>\breve{T}</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:		The median of the lognormal distribution, <math>\breve{T}\,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

	::<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 discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:		The mode of the lognormal distribution, <math>\tilde{T}\,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

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

	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math> , is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:		The standard deviation of the lognormal distribution, <math>{\sigma }_{T}\,\!</math>, is discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]:

	::<math>{\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)-\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}</math>		::<math>{\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)-\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\!</math>

	The standard deviation of the natural logarithms of the times-to-failure, <math>{\sigma}'\,\!</math> , in terms of <math>\bar{T}\,\!</math> and <math>{\sigma}\,\!</math> is given by:		The standard deviation of the natural logarithms of the times-to-failure, <math>{\sigma}'\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{\sigma}\,\!</math> is given by:

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

	===The Lognormal Reliability Function===		===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:		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(x)dx</math>		::<math>R(t)=\int_{t}^{\infty }f(x)dx\,\!</math>

	or:		or:

	::<math>{{R}({t})}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx</math>		::<math>{{R}({t})}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\!</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.
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	The lognormal conditional reliability function is given by:		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' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx}</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx}\,\!</math>

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

	===The Lognormal Reliable Life Function===		===The Lognormal Reliable Life Function===
	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 following equation for <math>t</math>:		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 following equation for <math>t\,\!</math>:

	::<math>{{R}_{t}}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx</math>		::<math>{{R}_{t}}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\!</math>

	===The Lognormal Failure Rate Function===		===The Lognormal Failure Rate Function===
	The lognormal failure rate is given by:		The lognormal failure rate is given by:

	::<math>\lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{t}'-{\mu }'}{{{\sigma' }}})}^{2}}}}}{\int_{{{t}'}}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{x-{\mu }'}{{{\sigma' }}})}^{2}}}}dx}</math>		::<math>\lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{t}'-{\mu }'}{{{\sigma' }}})}^{2}}}}}{\int_{{{t}'}}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{x-{\mu }'}{{{\sigma' }}})}^{2}}}}dx}\,\!</math>

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