Template:Lognormal distribution probability density function - Revision history

Richard House: Redirected page to The Lognormal Distribution

2012-08-13T04:41:09Z

Redirected page to The Lognormal Distribution

← Older revision		Revision as of 04:41, 13 August 2012
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	==Lognormal ~~Probability Density Function==~~		#REDIRECT [[The Lognormal Distribution]]

	~~The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:~~

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

	~~where,~~

	~~:<math>{t}'=\ln (t)</math>. <math>t</math> values are the times-to-failure, and~~

	~~:<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>~~

	~~:<math>\sigma'=\text{standard deviation of the natural logarithms of the times-to-failure}</math>~~

	~~The lognormal <math>pdf</math> can be obtained, realizing that for equal probabilities under the normal and lognormal <math>pdf</math> s, incremental areas should also be equal, or:~~

	~~::<math>f(t)dt=f({t}')d{t}'</math>~~

	~~Taking the derivative yields:~~

	~~::<math>d{t}'=\frac{dt}{t}</math>~~

	~~Substitution yields:~~

	~~::<math>\begin{align}~~
	~~f(t)= & \frac{f({t}')}{t}, \\~~
	~~f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}~~
	~~\end{align}</math>~~

	~~where:~~

	~~::<math>f(t)\ge 0,t>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>~~

Harry Guo at 16:13, 13 February 2012

2012-02-13T16:13:53Z

← Older revision		Revision as of 16:13, 13 February 2012
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	::<math>d{t}'=\frac{dt}{t}</math>		::<math>d{t}'=\frac{dt}{t}</math>

	:Substitution yields:		Substitution yields:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	:where:		where:

	::<math>f(t)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>		::<math>f(t)\ge 0,t>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>

Harry Guo at 16:13, 13 February 2012

2012-02-13T16:13:18Z

← Older revision		Revision as of 16:13, 13 February 2012
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	:<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>		:<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>

	:<math>\sigma'=\text{standard deviation of the natural logarithms~~}</math>~~		:<math>\sigma'=\text{standard deviation of the natural logarithms of the times-to-failure}</math>

	~~:<math>\text{~~of the times-to-failure}</math>

	The lognormal <math>pdf</math> can be obtained, realizing that for equal probabilities under the normal and lognormal <math>pdf</math> s, incremental areas should also be equal, or:		The lognormal <math>pdf</math> can be obtained, realizing that for equal probabilities under the normal and lognormal <math>pdf</math> s, incremental areas should also be equal, or:

Harry Guo at 16:13, 13 February 2012

2012-02-13T16:13:01Z

← Older revision		Revision as of 16:13, 13 February 2012
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	::<math>f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}</math>		::<math>f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}</math>

	where, <math>{t}'=\ln (t)</math>. ~~, where the~~ <math>t</math> values are the times-to-failure, and		where,

			:<math>{t}'=\ln (t)</math>. <math>t</math> values are the times-to-failure, and

	:<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>		:<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>

Harry Guo at 16:12, 13 February 2012

2012-02-13T16:12:27Z

← Older revision		Revision as of 16:12, 13 February 2012
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	where, <math>{t}'=\ln (t)</math>. , where the <math>t</math> values are the times-to-failure, and		where, <math>{t}'=\ln (t)</math>. , where the <math>t</math> values are the times-to-failure, and

	:<math>\mu'=\text{mean of the natural logarithms~~}</math>~~		:<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>

	~~:<math>\text{~~of the times-to-failure,}</math>

	:<math>\sigma'=\text{standard deviation of the natural logarithms}</math>		:<math>\sigma'=\text{standard deviation of the natural logarithms}</math>

Harry Guo at 16:11, 13 February 2012

2012-02-13T16:11:59Z

← Older revision		Revision as of 16:11, 13 February 2012
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	The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:		The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:

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

	where, <math>{T}'=\ln (T)</math>. , where the <math>T</math> values are the times-to-failure, and		where, <math>{t}'=\ln (t)</math>. , where the <math>t</math> values are the times-to-failure, and

	:<math>\mu'=\text{mean of the natural logarithms}</math>		:<math>\mu'=\text{mean of the natural logarithms}</math>
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	:<math>\text{of the times-to-failure,}</math>		:<math>\text{of the times-to-failure,}</math>

	:<math>\~~sigma_{T~~'}=\text{standard deviation of the natural logarithms}</math>		:<math>\sigma'=\text{standard deviation of the natural logarithms}</math>

	:<math>\text{of the times-to-failure}</math>		:<math>\text{of the times-to-failure}</math>
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	The lognormal <math>pdf</math> can be obtained, realizing that for equal probabilities under the normal and lognormal <math>pdf</math> s, incremental areas should also be equal, or:		The lognormal <math>pdf</math> can be obtained, realizing that for equal probabilities under the normal and lognormal <math>pdf</math> s, incremental areas should also be equal, or:

	::<math>f(T)dT=f({T}')d{T}'</math>		::<math>f(t)dt=f({t}')d{t}'</math>

	Taking the derivative yields:		Taking the derivative yields:

	::<math>d{T}'=\frac{dT}{T}</math>		::<math>d{t}'=\frac{dt}{t}</math>

	:Substitution yields:		:Substitution yields:

	::<math>\begin{align}		::<math>\begin{align}
	f(T)= & \frac{f({T}')}{T}, \\		f(t)= & \frac{f({t}')}{t}, \\
	f(T)= & \frac{1}{T\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(T)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}		f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}
	\end{align}</math>		\end{align}</math>

	:where:		:where:

	::<math>f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>		::<math>f(t)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>

Harry Guo at 16:10, 13 February 2012

2012-02-13T16:10:38Z

← Older revision		Revision as of 16:10, 13 February 2012
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	The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:		The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:

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

	where, <math>{T}'=\ln (T)</math>. , where the <math>T</math> values are the times-to-failure, and		where, <math>{T}'=\ln (T)</math>. , where the <math>T</math> values are the times-to-failure, and
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	::<math>\begin{align}		::<math>\begin{align}
	f(T)= & \frac{f({T}')}{T}, \\		f(T)= & \frac{f({T}')}{T}, \\
	f(T)= & \frac{1}{T\cdot {{\sigma ~~}_{{{T}~~'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(T)-{\mu }'}{{{\sigma ~~}_{{{T}~~'}}}} \right)}^{2}}}}		f(T)= & \frac{1}{T\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(T)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}
	\end{align}</math>		\end{align}</math>

	:where:		:where:

	::<math>f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma ~~}_{{{T}~~'}}}>0</math>		::<math>f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>

Harry Guo at 16:09, 13 February 2012

2012-02-13T16:09:45Z

← Older revision		Revision as of 16:09, 13 February 2012
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	The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:		The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:

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

	where, <math>{T}'=\ln (T)</math>. , where the <math>T</math> values are the times-to-failure, and		where, <math>{T}'=\ln (T)</math>. , where the <math>T</math> values are the times-to-failure, and

Harry Guo at 16:08, 13 February 2012

2012-02-13T16:08:12Z

← Older revision		Revision as of 16:08, 13 February 2012
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	==Lognormal Probability Density Function==		==Lognormal Probability Density Function==

	The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>{{\sigma }'}}</math> . The <math>pdf</math> for this distribution is given by:		The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>\sigma'</math> . The <math>pdf</math> for this distribution is given by:

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

Harry Guo at 16:07, 13 February 2012

2012-02-13T16:07:41Z

← Older revision		Revision as of 16:07, 13 February 2012
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	==Lognormal Probability Density Function==		==Lognormal Probability Density Function==

	The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>{{\sigma }'}}}</math> . The <math>pdf</math> for this distribution is given by:		The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>{{\sigma }'}}</math> . The <math>pdf</math> for this distribution is given by:

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