Lognormal Distribution Functions: Difference between revisions
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<noinclude>{{Navigation box}}[[Category:Shared Articles]] | <noinclude>{{Navigation box}}[[Category:Shared Articles]] | ||
''This article also appears in the [ | ''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, | 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 | 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, | 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, | 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, | 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) | ::<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, | 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 | 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. | ||
Latest revision as of 21:43, 18 September 2023
This article also appears in the Life data analysis reference and Accelerated life testing reference.
The Mean or MTTF
The mean of the lognormal distribution, [math]\displaystyle{ \mu \,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}\,\! }[/math]
The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ \mu'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {{\sigma}}\,\! }[/math] is given by:
- [math]\displaystyle{ {\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\! }[/math]
The Median
The median of the lognormal distribution, [math]\displaystyle{ \breve{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \breve{T}={{e}^{{{\mu}'}}}\,\! }[/math]
The Mode
The mode of the lognormal distribution, [math]\displaystyle{ \tilde{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \tilde{T}={{e}^{{\mu }'-\sigma'^{2}}}\,\! }[/math]
The Standard Deviation
The standard deviation of the lognormal distribution, [math]\displaystyle{ {\sigma }_{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ {\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]\displaystyle{ {\sigma}'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {\sigma}\,\! }[/math] is given by:
- [math]\displaystyle{ \sigma '=\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]\displaystyle{ t\,\! }[/math], starting at age 0, for the lognormal distribution is determined by:
- [math]\displaystyle{ R(t)=\int_{t}^{\infty }f(x)dx\,\! }[/math]
or:
- [math]\displaystyle{ {{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.
The Lognormal Conditional Reliability Function
The lognormal conditional reliability function is given by:
- [math]\displaystyle{ 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.
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]\displaystyle{ t\,\! }[/math]:
- [math]\displaystyle{ {{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 is given by:
- [math]\displaystyle{ \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.