Template:Lognormal distribution probability plotting - Revision history

Richard House: Redirected page to The Lognormal Distribution

2012-08-13T05:41:16Z

Redirected page to The Lognormal Distribution

← Older revision		Revision as of 05:41, 13 August 2012
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	#REDIRECT [[~~The_Loglogistic_Distribution~~]]		#REDIRECT [[The_Lognormal_Distribution]]

Lisa Hacker: Redirected page to The Loglogistic Distribution

2012-08-09T10:02:02Z

Redirected page to The Loglogistic Distribution

← Older revision		Revision as of 10:02, 9 August 2012
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	~~===Probability Plotting===~~		#REDIRECT [[The_Loglogistic_Distribution]]
	As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the <math>cdf</math> of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:

	~~::<math>F({t}')=\Phi \left( \frac{{t}'-{\mu }'}{{{\sigma'}}} \right)</math>~~

	~~or:~~

	~~::<math>{{\Phi }^{-1}}\left[ F({t}') \right]=-\frac{{{\mu }'}}{{{\sigma}'}}+\frac{1}{{{\sigma }'}}\cdot {t}'</math>~~

	~~where:~~

	~~::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>~~

	~~Now, let:~~

	~~::<math>y={{\Phi }^{-1}}\left[ F({t}') \right]</math>~~

	~~::<math>a=-\frac{{{\mu }'}}{{{\sigma}'}}</math>~~

	~~and:~~

	~~::<math>b=\frac{1}{{{\sigma}'}}</math>~~

	~~which results in the linear equation of:~~

	~~::<math>y=a+b{t}'</math>~~

	~~The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.~~

	~~[[Image:BS.10 lognormal probability plot.png\|center\|thumb\|250px\| ]]~~

	~~The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see~~ [[~~The Normal Distribution~~]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

	~~The process of lognormal probability plotting is illustrated in the following example.~~


	~~'''Example 1:'''~~
	~~{{Example: Lognormal Distribution Probability Plot}}~~

Nicolette Young: /* Probability Plotting */

2012-04-25T21:19:52Z

Probability Plotting

← Older revision		Revision as of 21:19, 25 April 2012
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	The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.		The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.

	[[Image:BS.10 lognormal probability plot.png\|center\|250px\| ]]		[[Image:BS.10 lognormal probability plot.png\|center\|thumb\|250px\| ]]

	The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.		The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

Nicolette Young: /* Probability Plotting */

2012-04-25T21:18:42Z

Probability Plotting

← Older revision		Revision as of 21:18, 25 April 2012
Line 28:		Line 28:
	The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.		The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.

	[[Image:BS.10 lognormal probability plot.png\|center\|~~400px~~\| ]]		[[Image:BS.10 lognormal probability plot.png\|center\|250px\| ]]

	The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.		The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

Nicolette Young: /* Probability Plotting */

2012-03-14T19:08:05Z

Probability Plotting

← Older revision		Revision as of 19:08, 14 March 2012
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	The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.		The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.

	[[Image:~~lda_lognormalplot~~.~~gif\|thumb~~\|center\|400px\| ]]		[[Image:BS.10 lognormal probability plot.png\|center\|400px\| ]]

	The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.		The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

Harry Guo: /* Probability Plotting */

2012-02-13T21:01:17Z

Probability Plotting

← Older revision		Revision as of 21:01, 13 February 2012
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	The process of lognormal probability plotting is illustrated in the following example.		The process of lognormal probability plotting is illustrated in the following example.


	'''Example 1:'''		'''Example 1:'''
	{{Example: Lognormal Distribution Probability Plot}}		{{Example: Lognormal Distribution Probability Plot}}

Harry Guo at 18:12, 13 February 2012

2012-02-13T18:12:20Z

← Older revision		Revision as of 18:12, 13 February 2012
Line 34:		Line 34:
	The process of lognormal probability plotting is illustrated in the following example.		The process of lognormal probability plotting is illustrated in the following example.

	{{Example: Lognormal Distribution}}		'''Example 1:'''
			{{Example: Lognormal Distribution Probability Plot}}

Harry Guo: /* Probability Plotting */

2012-02-13T17:41:36Z

Probability Plotting

← Older revision		Revision as of 17:41, 13 February 2012
Line 30:		Line 30:
	[[Image:lda_lognormalplot.gif\|thumb\|center\|400px\| ]]		[[Image:lda_lognormalplot.gif\|thumb\|center\|400px\| ]]

	The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.		The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

	The process of lognormal probability plotting is illustrated in the following example.		The process of lognormal probability plotting is illustrated in the following example.

	{{Example: Lognormal Distribution}}		{{Example: Lognormal Distribution}}

Harry Guo: /* Probability Plotting */

2012-02-13T17:40:46Z

Probability Plotting

← Older revision		Revision as of 17:40, 13 February 2012
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	where:		where:

	::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dx</math>		::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

	Now, let:		Now, let:

Harry Guo: /* Probability Plotting */

2012-02-13T17:40:14Z

Probability Plotting

← Older revision		Revision as of 17:40, 13 February 2012
Line 2:		Line 2:
	As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the <math>cdf</math> of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:		As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the <math>cdf</math> of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:

	::<math>F({T}')=\Phi \left( \frac{{T}'-{\mu }'}{{{\sigma ~~}_{{{T}~~'}}}} \right)</math>		::<math>F({t}')=\Phi \left( \frac{{t}'-{\mu }'}{{{\sigma'}}} \right)</math>

	:or:		or:

	::<math>{{\Phi }^{-1}}\left[ F({T}') \right]=-\frac{{{\mu }'}}{{{\sigma ~~}_{{{T~~}'}}}}+\frac{1}{{{\sigma ~~}_{{{T~~}'}}}}\cdot {T}'</math>		::<math>{{\Phi }^{-1}}\left[ F({t}') \right]=-\frac{{{\mu }'}}{{{\sigma}'}}+\frac{1}{{{\sigma }'}}\cdot {t}'</math>

	:where:		where:

	::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dx</math>		::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dx</math>

	:Now, let:		Now, let:

	::<math>y={{\Phi }^{-1}}\left[ F({T}') \right]</math>		::<math>y={{\Phi }^{-1}}\left[ F({t}') \right]</math>

	::<math>a=-\frac{{{\mu }'}}{{{\sigma ~~}_{{{T~~}'}}}}</math>		::<math>a=-\frac{{{\mu }'}}{{{\sigma}'}}</math>

	:and:		and:

	::<math>b=\frac{1}{{{\sigma ~~}_{{{T~~}'}}}}</math>		::<math>b=\frac{1}{{{\sigma}'}}</math>

	which results in the linear equation of:		which results in the linear equation of:

	::<math>y=a+b{T}'</math>		::<math>y=a+b{t}'</math>

	The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.		The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.
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	[[Image:lda_lognormalplot.gif\|thumb\|center\|400px\| ]]		[[Image:lda_lognormalplot.gif\|thumb\|center\|400px\| ]]

	The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see Chapter 8). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.		The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

	The process of lognormal probability plotting is illustrated in the following example.		The process of lognormal probability plotting is illustrated in the following example.

	{{Example: Lognormal Distribution}}		{{Example: Lognormal Distribution}}