The Logistic Distribution - Revision history

Lisa Hacker at 00:59, 10 March 2023

2023-03-10T00:59:25Z

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 For  <math>z=1\,\!</math>, <math>\sigma =t-\mu \,\!</math> and <math>R(t)=\tfrac{1}{1+\exp (1)}\approx 0.2689.\,\!</math> Therefore, <math>\sigma \,\!</math> can be found by subtracting <math>\mu \,\!</math> from the time value where the plotted probability line crosses the 73.10% unreliability (26.89% reliability) horizontal line.
 ==General Example==
 {{:Logistic Distribution Example}}

Lisa Hacker at 00:52, 10 March 2023

2023-03-10T00:52:00Z

← Older revision		Revision as of 00:52, 10 March 2023
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	If <math>\delta \,\!</math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}\,\!</math> for the two-sided bounds, and <math>\alpha =1-\delta \,\!</math> for the one-sided bounds.		If <math>\delta \,\!</math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}\,\!</math> for the two-sided bounds, and <math>\alpha =1-\delta \,\!</math> for the one-sided bounds.

	The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{\sigma }\,\!</math> are estimated from the Fisher matrix, as follows:		<!--The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{\sigma }\,\!</math> are estimated from the Fisher matrix, as follows:

	::<math>\left( \begin{matrix}		::<math>\left( \begin{matrix}
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	<math>\Lambda \,\!</math> is the log-likelihood function of the normal distribution, described in [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations\|Appendix D]].		<math>\Lambda \,\!</math> is the log-likelihood function of the normal distribution, described in [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations\|Appendix D]].-->

	===Bounds on Reliability===		===Bounds on Reliability===

Lisa Hacker: /* Logistic Probability Density Function */

2023-03-10T00:48:57Z

Logistic Probability Density Function

← Older revision		Revision as of 00:48, 10 March 2023
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	The logistic ''pdf'' is given by:		The logistic ''pdf'' is given by:

	::<math>\		::<math>\begin{align}
	f(t)=\tfrac{{{e}^{z}}}{\sigma {{(1+{{e}^{z}})}^{2}}} \\ \\		f(t)=\tfrac{{{e}^{z}}}{\sigma {{(1+{{e}^{z}})}^{2}}} \\ \\

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	-\infty <t<\infty ,\ \ -\infty <\mu <\infty ,\sigma >0 \\ \\		-\infty <t<\infty ,\ \ -\infty <\mu <\infty ,\sigma >0 \\ \\
	\,\!</math>		\end{align}\,\!</math>

	where:		where:

Lisa Hacker at 00:47, 10 March 2023

2023-03-10T00:47:16Z

← Older revision		Revision as of 00:47, 10 March 2023
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	The logistic ''pdf'' is given by:		The logistic ''pdf'' is given by:

	::<math>\~~begin{matrix}~~		::<math>\
	f(t)=\tfrac{{{e}^{z}}}{\sigma {{(1+{{e}^{z}})}^{2}}} \\ \\		f(t)=\tfrac{{{e}^{z}}}{\sigma {{(1+{{e}^{z}})}^{2}}} \\ \\

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	-\infty <t<\infty ,\ \ -\infty <\mu <\infty ,\sigma >0 \\ \\		-\infty <t<\infty ,\ \ -\infty <\mu <\infty ,\sigma >0 \\ \\
	~~\end{matrix}~~\,\!</math>		\,\!</math>

	where:		where:

Nicolette Young: /* Logistic Distribution Probability Paper */

2015-12-23T17:10:59Z

Logistic Distribution Probability Paper

← Older revision		Revision as of 17:10, 23 December 2015
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	The logistic probability paper resulting from this linearized ''cdf'' function is shown next.		The logistic probability paper resulting from this linearized ''cdf'' function is shown next.

	[[Image:WB.14 logistic probability plot.png\|center\|~~350px~~\| ]]		[[Image:WB.14 logistic probability plot.png\|center\|400px\| ]]

	Since the logistic distribution is symmetrical, the area under the ''pdf'' curve from <math>-\infty \,\!</math> to <math>\mu \,\!</math> is 0.5, as is the area from <math>\mu \,\!</math> to <math>+\infty \,\!</math>. Consequently, the value of <math>\mu \,\!</math> is said to be the point where <math>R(t)=Q(t)=50%\,\!</math>. This means that the estimate of <math>\mu \,\!</math> can be read from the point where the plotted line crosses the 50% unreliability line.		Since the logistic distribution is symmetrical, the area under the ''pdf'' curve from <math>-\infty \,\!</math> to <math>\mu \,\!</math> is 0.5, as is the area from <math>\mu \,\!</math> to <math>+\infty \,\!</math>. Consequently, the value of <math>\mu \,\!</math> is said to be the point where <math>R(t)=Q(t)=50%\,\!</math>. This means that the estimate of <math>\mu \,\!</math> can be read from the point where the plotted line crosses the 50% unreliability line.

Nicolette Young: /* Characteristics of the Logistic Distribution */

2015-12-23T17:10:47Z

Characteristics of the Logistic Distribution

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	:* As <math>\mu \,\!</math> increases, the ''pdf'' is shifted to the right.		:* As <math>\mu \,\!</math> increases, the ''pdf'' is shifted to the right.

	[[Image:WB.14 effect of mu.png\|center\|~~350px~~\| ]]		[[Image:WB.14 effect of mu.png\|center\|400px\| ]]

	:* As <math>\sigma \,\!</math> decreases, the ''pdf'' gets pushed toward the mean, or it becomes narrower and taller.		:* As <math>\sigma \,\!</math> decreases, the ''pdf'' gets pushed toward the mean, or it becomes narrower and taller.
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	:* The scale parameter can assume values of <math>0<\sigma <\infty \,\!</math>.		:* The scale parameter can assume values of <math>0<\sigma <\infty \,\!</math>.

	[[Image:WB.14 effect of sigma.png\|center\|~~350px~~\| ]]		[[Image:WB.14 effect of sigma.png\|center\|400px\| ]]

	:* The logistic ''pdf'' starts at <math>t=-\infty \,\!</math> with an <math>f(t)=0\,\!</math>. As <math>t\,\!</math> increases, <math>f(t)\,\!</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}\,\!</math>. Thereafter, <math>f(t)\,\!</math> decreases, goes through its point of inflection and assumes a value of <math>f(t)=0\,\!</math> at <math>t=+\infty \,\!</math>.		:* The logistic ''pdf'' starts at <math>t=-\infty \,\!</math> with an <math>f(t)=0\,\!</math>. As <math>t\,\!</math> increases, <math>f(t)\,\!</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}\,\!</math>. Thereafter, <math>f(t)\,\!</math> decreases, goes through its point of inflection and assumes a value of <math>f(t)=0\,\!</math> at <math>t=+\infty \,\!</math>.

Nicolette Young: /* The Logistic Mean, Median and Mode */

2015-12-23T17:10:30Z

The Logistic Mean, Median and Mode

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	the median and the mode are always equal to the mean, <math>\mu =\tilde{T}=\breve{T}.\,\!</math>		the median and the mode are always equal to the mean, <math>\mu =\tilde{T}=\breve{T}.\,\!</math>

	[[Image:WB.14 logistic pd.png\|center\|~~250px~~\| ]]		[[Image:WB.14 logistic pd.png\|center\|400px\| ]]

	===The Logistic Standard Deviation===		===The Logistic Standard Deviation===

Lisa Hacker at 23:55, 23 June 2015

2015-06-23T23:55:10Z

← Older revision		Revision as of 23:55, 23 June 2015
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	{{:Logistic Distribution Example}}		{{:Logistic Distribution Example}}


	~~[[Category: Completed Theoretical Review]]~~

Richard House at 16:27, 26 September 2012

2012-09-26T16:27:13Z

Show changes

Richard House at 17:45, 5 September 2012

2012-09-05T17:45:18Z

← Older revision		Revision as of 17:45, 5 September 2012
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	===Logistic Probability Density Function===		===Logistic Probability Density Function===
	The logistic ~~<math>~~pdf~~</math>~~ is given by:		The logistic ''pdf'' is given by:

	::<math>\begin{matrix}		::<math>\begin{matrix}
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	==Characteristics of the Logistic Distribution==		==Characteristics of the Logistic Distribution==
	:* The logistic distribution has no shape parameter. This means that the logistic ~~<math>~~pdf~~</math>~~ has only one shape, the bell shape, and this shape does not change. The shape of the logistic distribution is very similar to that of the normal distribution.		:* The logistic distribution has no shape parameter. This means that the logistic ''pdf'' has only one shape, the bell shape, and this shape does not change. The shape of the logistic distribution is very similar to that of the normal distribution.
	:* The mean, <math>\mu \,\!</math> , or the mean life or the <math>MTTF\,\!</math> , is also the location parameter of the logistic ~~<math>~~pdf~~</math>~~ , as it locates the ~~<math>~~pdf~~</math>~~ along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty \,\!</math> .		:* The mean, <math>\mu \,\!</math> , or the mean life or the <math>MTTF\,\!</math> , is also the location parameter of the logistic ''pdf'', as it locates the ''pdf'' along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty \,\!</math> .
	:* As <math>\mu </math> decreases, the ~~<math>~~pdf~~</math>~~ is shifted to the left.		:* As <math>\mu </math> decreases, the ''pdf'' is shifted to the left.
	:* As <math>\mu </math> increases, the ~~<math>~~pdf~~</math>~~ is shifted to the right.		:* As <math>\mu </math> increases, the ''pdf'' is shifted to the right.

	[[Image:WB.14 effect of mu.png\|center\|350px\| ]]		[[Image:WB.14 effect of mu.png\|center\|350px\| ]]

	:* As <math>\sigma \,\!</math> decreases, the ~~<math>~~pdf~~</math>~~ gets pushed toward the mean, or it becomes narrower and taller.		:* As <math>\sigma \,\!</math> decreases, the ''pdf'' gets pushed toward the mean, or it becomes narrower and taller.
	:* As <math>\sigma \,\!</math> increases, the ~~<math>~~pdf~~</math>~~ spreads out away from the mean, or it becomes broader and shallower.		:* As <math>\sigma \,\!</math> increases, the ''pdf'' spreads out away from the mean, or it becomes broader and shallower.
	:* The scale parameter can assume values of <math>0<\sigma <\infty \,\!</math>.		:* The scale parameter can assume values of <math>0<\sigma <\infty \,\!</math>.

	[[Image:WB.14 effect of sigma.png\|center\|350px\| ]]		[[Image:WB.14 effect of sigma.png\|center\|350px\| ]]

	:* The logistic ~~<math>~~pdf~~</math>~~ starts at <math>t=-\infty \,\!</math> with an <math>f(t)=0\,\!</math> . As <math>t</math> increases, <math>f(t)\,\!</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}\,\!</math> . Thereafter, <math>f(t)\,\!</math> decreases, goes through its point of inflection and assumes a value of <math>f(t)=0\,\!</math> at <math>t=+\infty \,\!</math> .		:* The logistic ''pdf'' starts at <math>t=-\infty \,\!</math> with an <math>f(t)=0\,\!</math> . As <math>t</math> increases, <math>f(t)\,\!</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}\,\!</math> . Thereafter, <math>f(t)\,\!</math> decreases, goes through its point of inflection and assumes a value of <math>f(t)=0\,\!</math> at <math>t=+\infty \,\!</math> .
	:* For <math>t=\pm \infty ,\,\!</math> the ~~<math>~~pdf~~</math>~~ equals 0. The maximum value of the ~~<math>~~pdf~~</math>~~ occurs at <math>t = \mu \,\!</math> and equals <math>\tfrac{1}{4\sigma }.\,\!</math>		:* For <math>t=\pm \infty ,\,\!</math> the ''pdf'' equals 0. The maximum value of the ''pdf'' occurs at <math>t = \mu \,\!</math> and equals <math>\tfrac{1}{4\sigma }.\,\!</math>
	:* The point of inflection of the ~~<math>~~pdf~~</math>~~ plot is the point where the second derivative of the ~~<math>~~pdf~~</math>~~ equals zero. The inflection point occurs at <math>t=\mu +\sigma \ln (2\pm \sqrt{3})\,\!</math> or <math>t\approx \mu \pm \sigma 1.31696\,\!</math>.		:* The point of inflection of the ''pdf'' plot is the point where the second derivative of the ''pdf'' equals zero. The inflection point occurs at <math>t=\mu +\sigma \ln (2\pm \sqrt{3})\,\!</math> or <math>t\approx \mu \pm \sigma 1.31696\,\!</math>.
	:* If the location parameter <math>\mu \,\!</math> decreases, the reliability plot is shifted to the left. If <math>\mu \,\!</math> increases, the reliability plot is shifted to the right.		:* If the location parameter <math>\mu \,\!</math> decreases, the reliability plot is shifted to the left. If <math>\mu \,\!</math> increases, the reliability plot is shifted to the right.
	:* If <math>t=\mu \,\!</math> then <math>R=0.5\,\!</math> . is the inflection point. If <math>t<\mu \,\!</math> then <math>R(t)\,\!</math> is concave (concave down); if <math>t>\mu \,\!</math> then <math>R(t)\,\!</math> is convex (concave up). For <math>t<\mu ,\,\!</math> <math>\lambda (t)\,\!</math> is convex (concave up), for <math>t>\mu ;\,\!</math> <math>\lambda (t)\,\!</math> is concave (concave down).		:* If <math>t=\mu \,\!</math> then <math>R=0.5\,\!</math> . is the inflection point. If <math>t<\mu \,\!</math> then <math>R(t)\,\!</math> is concave (concave down); if <math>t>\mu \,\!</math> then <math>R(t)\,\!</math> is convex (concave up). For <math>t<\mu ,\,\!</math> <math>\lambda (t)\,\!</math> is convex (concave up), for <math>t>\mu ;\,\!</math> <math>\lambda (t)\,\!</math> is concave (concave down).
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	==Logistic Distribution Probability Paper==		==Logistic Distribution Probability Paper==
	The form of the Logistic probability paper is based on linearizing the ~~<math>~~cdf~~</math>~~ .		The form of the Logistic probability paper is based on linearizing the ''cdf''.
	From unreliability equation, <math>z\,\!</math> can be calculated as a function of the ~~<math>cdf</math>~~ <math>F</math> as follows:		From unreliability equation, <math>z\,\!</math> can be calculated as a function of the ''cdf'' <math>F</math> as follows:

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

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

	[[Image:WB.14 logistic probability plot.png\|center\|350px\| ]]		[[Image:WB.14 logistic probability plot.png\|center\|350px\| ]]

	Since the logistic distribution is symmetrical, the area under the ~~<math>~~pdf~~</math>~~ curve from <math>-\infty \,\!</math> to <math>\mu \,\!</math> is 0.5, as is the area from <math>\mu \,\!</math> to <math>+\infty \,\!</math> . Consequently, the value of <math>\mu \,\!</math> is said to be the point where <math>R(t)=Q(t)=50%\,\!</math> . This means that the estimate of <math>\mu \,\!</math> can be read from the point where the plotted line crosses the 50% unreliability line.		Since the logistic distribution is symmetrical, the area under the ''pdf'' curve from <math>-\infty \,\!</math> to <math>\mu \,\!</math> is 0.5, as is the area from <math>\mu \,\!</math> to <math>+\infty \,\!</math> . Consequently, the value of <math>\mu \,\!</math> is said to be the point where <math>R(t)=Q(t)=50%\,\!</math> . This means that the estimate of <math>\mu \,\!</math> can be read from the point where the plotted line crosses the 50% unreliability line.

	For <math>z=1\,\!</math> , <math>\sigma =t-\mu \,\!</math> and <math>R(t)=\tfrac{1}{1+\exp (1)}\approx 0.2689.\,\!</math> Therefore, <math>\sigma \,\!</math> can be found by subtracting <math>\mu \,\!</math> from the time value where the plotted probability line crosses the 73.10% unreliability (26.89% reliability) horizontal line.		For <math>z=1\,\!</math> , <math>\sigma =t-\mu \,\!</math> and <math>R(t)=\tfrac{1}{1+\exp (1)}\approx 0.2689.\,\!</math> Therefore, <math>\sigma \,\!</math> can be found by subtracting <math>\mu \,\!</math> from the time value where the plotted probability line crosses the 73.10% unreliability (26.89% reliability) horizontal line.