Lognormal Statistical Properties: Difference between revisions
Steve Sharp (talk | contribs) (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…') |
Steve Sharp (talk | contribs) No edit summary |
||
| Line 2: | Line 2: | ||
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> | ||
| Line 33: | Line 33: | ||
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. | ||
Revision as of 17:13, 29 June 2011
The Mean or MTTF
The mean of the lognormal distribution, [math]\displaystyle{ \mu }[/math] , is given by [18]:
- [math]\displaystyle{ \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{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 }_{T}} }[/math] is givgen by:
- [math]\displaystyle{ {\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]\displaystyle{ \breve{T} }[/math] , is given by [18]:
- [math]\displaystyle{ \breve{T}={{e}^{{{\mu }'}}} }[/math]
The Mode
The mode of the lognormal distribution, [math]\displaystyle{ \tilde{T} }[/math] , is given by [1]:
- [math]\displaystyle{ \tilde{T}={{e}^{{\mu }'-\sigma _{{{T}'}}^{2}}} }[/math]
The Standard Deviation
The standard deviation of the lognormal distribution, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by [18]:
- [math]\displaystyle{ {{\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]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:
- [math]\displaystyle{ {{\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]\displaystyle{ T }[/math] , starting at age 0, for the lognormal distribution is determined by:
- [math]\displaystyle{ R(T)=\int_{T}^{\infty }f(t)dt }[/math]
or:
- [math]\displaystyle{ 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.