The Gamma Distribution - Revision history

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

2015-12-23T17:10:11Z

Characteristics of the Gamma Distribution

← Older revision		Revision as of 17:10, 23 December 2015
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	:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>		:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>

	[[Image:BSpdf1.png\|center\|~~350px~~\| ]]		[[Image:BSpdf1.png\|center\|400px\| ]]

	For <math>k=1\,\!</math> :		For <math>k=1\,\!</math> :
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	:• The mode does not exist.		:• The mode does not exist.

	[[Image:BSpdf2.png\|center\|~~350px~~\| ]]		[[Image:BSpdf2.png\|center\|400px\| ]]

	For <math>0<k<1\,\!</math> :		For <math>0<k<1\,\!</math> :
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	:• The mode does not exist.		:• The mode does not exist.

	[[Image:BSpdf3.png\|center\|~~350px~~\| ]]		[[Image:BSpdf3.png\|center\|400px\| ]]

	==Confidence Bounds==		==Confidence Bounds==

Richard House at 16:20, 26 September 2012

2012-09-26T16:20:51Z

Show changes

Richard House at 16:34, 5 September 2012

2012-09-05T16:34:58Z

← Older revision		Revision as of 16:34, 5 September 2012
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	{{template:LDABOOK\|13\|The Gamma Distribution}}		{{template:LDABOOK\|13\|The Gamma Distribution}}
	The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications [[Appendix: ~~Weibull References~~\|[32]]].		The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications, as discussed in [[Appendix:_Life_Data_Analysis_References\|[32]]].

	===The Gamma Probability Density Function===		===The Gamma Probability Density Function===

Richard House: /* Characteristics of the Gamma Distribution */

2012-09-04T22:57:46Z

Characteristics of the Gamma Distribution

Richard House: /* The Gamma Probability Density Function */

2012-09-04T22:56:30Z

The Gamma Probability Density Function

← Older revision		Revision as of 22:56, 4 September 2012
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	===The Gamma Probability Density Function===		===The Gamma Probability Density Function===
	The ~~<math>~~pdf~~</math>~~ of the gamma distribution is given by:		The ''pdf'' of the gamma distribution is given by:

	::<math>f(t)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)}</math>		::<math>f(t)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)}</math>

Richard House: /* Bounds on the Parameters */

2012-08-28T03:08:47Z

Bounds on the Parameters

← Older revision		Revision as of 03:08, 28 August 2012
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	\end{align}</math>		\end{align}</math>

	where <math>{{K}_{\alpha }}</math> is defined by:		where <math>{{K}_{\alpha }}\,\!</math> is defined by:

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

	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{k}</math> are estimated from the Fisher matrix, as follows:		The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{k}\,\!</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 gamma distribution, described in [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations\|Appendix D]]		<math>\Lambda \,\!</math> is the log-likelihood function of the gamma distribution, described in [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations\|Appendix D]]


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

Richard House: /* Characteristics of the Gamma Distribution */

2012-08-28T03:08:00Z

Characteristics of the Gamma Distribution

← Older revision		Revision as of 03:08, 28 August 2012
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	Some of the specific characteristics of the gamma distribution are the following:		Some of the specific characteristics of the gamma distribution are the following:

	For <math>k>1</math> :		For <math>k>1\,\!</math> :
	:• As <math>t\to 0,\infty </math> , <math>f(t)\to 0.</math>		:• As <math>t\to 0,\infty\,\! </math> , <math>f(t)\to 0.\,\!</math>
	:• <math>f(t)</math> increases from 0 to the mode value and decreases thereafter.		:• <math>f(t)\,\!</math> increases from 0 to the mode value and decreases thereafter.
	:• If <math>k\le 2</math> then <math>pdf</math> has one inflection point at <math>t={{e}^{\mu }}\sqrt{k-1}(</math> <math>\sqrt{k-1}+1).</math>		:• If <math>k\le 2\,\!</math> then <math>pdf</math> has one inflection point at <math>t={{e}^{\mu }}\sqrt{k-1}(</math> <math>\sqrt{k-1}+1).\,\!</math>
	:• If <math>k>2</math> then <math>pdf</math> has two inflection points for <math>t={{e}^{\mu }}\sqrt{k-1}(</math> <math>\sqrt{k-1}\pm 1).</math>		:• If <math>k>2\,\!</math> then <math>pdf</math> has two inflection points for <math>t={{e}^{\mu }}\sqrt{k-1}(\,\!</math> <math>\sqrt{k-1}\pm 1).\,\!</math>
	:• For a fixed <math>k</math> , as <math>\mu </math> increases, the <math>pdf</math> starts to look more like a straight angle.		:• For a fixed <math>k\,\!</math> , as <math>\mu \,\!</math> increases, the <math>pdf</math> starts to look more like a straight angle.
	:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.</math>		:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>

	[[Image:BSpdf1.png\|center\|350px\| ]]		[[Image:BSpdf1.png\|center\|350px\| ]]
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	For <math>k=1</math> :		For <math>k=1</math> :
	:• Gamma becomes the exponential distribution.		:• Gamma becomes the exponential distribution.
	:• As <math>t\to 0</math> , <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.</math>		:• As <math>t\to 0\,\!</math> , <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>
	:• As <math>t\to \infty ,f(t)\to 0.</math>		:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>
	:• The <math>pdf</math> decreases monotonically and is convex.		:• The <math>pdf</math> decreases monotonically and is convex.
	:• <math>\lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}</math> . <math>\lambda (t)</math> is constant.		:• <math>\lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}\,\!</math> . <math>\lambda (t)\,\!</math> is constant.
	:• The mode does not exist.		:• The mode does not exist.

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	For <math>0<k<1</math> :		For <math>0<k<1</math> :
	:• As <math>t\to 0</math> , <math>f(t)\to \infty .</math>		:• As <math>t\to 0\,\!</math> , <math>f(t)\to \infty .\,\!</math>
	:• As <math>t\to \infty ,f(t)\to 0.</math>		:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>
	:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.</math>		:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>
	:• The <math>pdf</math> decreases monotonically and is convex.		:• The <math>pdf\,\!</math> decreases monotonically and is convex.
	:• As <math>\mu </math> increases, the <math>pdf</math> gets stretched out to the right and its height decreases, while maintaining its shape.		:• As <math>\mu \,\!</math> increases, the <math>pdf</math> gets stretched out to the right and its height decreases, while maintaining its shape.
	:• As <math>\mu </math> decreases, the <math>pdf</math> shifts towards the left and its height increases.		:• As <math>\mu \,\!</math> decreases, the <math>pdf</math> shifts towards the left and its height increases.
	:• The mode does not exist.		:• The mode does not exist.

Richard House: /* The Gamma Mean, Median and Mode */

2012-08-28T02:46:13Z

The Gamma Mean, Median and Mode

← Older revision		Revision as of 02:46, 28 August 2012
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	::<math>\overline{T}=k{{e}^{\mu }}</math>		::<math>\overline{T}=k{{e}^{\mu }}</math>

	The mode exists if <math>k>1</math> and is given by:		The mode exists if <math>k>1\,\!</math> and is given by:

	::<math>\tilde{T}=(k-1){{e}^{\mu }}</math>		::<math>\tilde{T}=(k-1){{e}^{\mu }}</math>

Richard House: /* The Gamma Reliability Function */

2012-08-28T02:45:56Z

The Gamma Reliability Function

← Older revision		Revision as of 02:45, 28 August 2012
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	===The Gamma Reliability Function===		===The Gamma Reliability Function===
	The reliability for a mission of time <math>t</math> for the gamma distribution is:		The reliability for a mission of time <math>t\,\!</math> for the gamma distribution is:

	::<math>R=1-{{\Gamma }_{I}}(k;{{e}^{z}})</math>		::<math>\begin{align}
			R=1-{{\Gamma }_{I}}(k;{{e}^{z}})
			\end{align}</math>

	===The Gamma Mean, Median and Mode===		===The Gamma Mean, Median and Mode===

Chris Kahn: /* General Example */

2012-08-20T01:09:15Z

General Example

← Older revision		Revision as of 01:09, 20 August 2012
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	==General Example==		==General Example==
	{{~~:Example~~: Gamma Distribution}}		{{:Gamma Distribution Example}}

← Older revision		Revision as of 22:57, 4 September 2012
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	:• As <math>t\to 0,\infty\,\! </math> , <math>f(t)\to 0.\,\!</math>		:• As <math>t\to 0,\infty\,\! </math> , <math>f(t)\to 0.\,\!</math>
	:• <math>f(t)\,\!</math> increases from 0 to the mode value and decreases thereafter.		:• <math>f(t)\,\!</math> increases from 0 to the mode value and decreases thereafter.
	:• If <math>k\le 2\,\!</math> then ~~<math>~~pdf~~</math>~~ has one inflection point at <math>t={{e}^{\mu }}\sqrt{k-1}(</math> <math>\sqrt{k-1}+1).\,\!</math>		:• If <math>k\le 2\,\!</math> then ''pdf'' has one inflection point at <math>t={{e}^{\mu }}\sqrt{k-1}(</math> <math>\sqrt{k-1}+1).\,\!</math>
	:• If <math>k>2\,\!</math> then ~~<math>~~pdf~~</math>~~ has two inflection points for <math>t={{e}^{\mu }}\sqrt{k-1}(\,\!</math> <math>\sqrt{k-1}\pm 1).\,\!</math>		:• If <math>k>2\,\!</math> then ''pdf'' has two inflection points for <math>t={{e}^{\mu }}\sqrt{k-1}(\,\!</math> <math>\sqrt{k-1}\pm 1).\,\!</math>
	:• For a fixed <math>k\,\!</math> , as <math>\mu \,\!</math> increases, the ~~<math>~~pdf~~</math>~~ starts to look more like a straight angle.		:• For a fixed <math>k\,\!</math> , as <math>\mu \,\!</math> increases, the ''pdf'' starts to look more like a straight angle.
	:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>		:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>

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	:• As <math>t\to 0\,\!</math> , <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>		:• As <math>t\to 0\,\!</math> , <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>
	:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>		:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>
	:• The ~~<math>~~pdf~~</math>~~ decreases monotonically and is convex.		:• The ''pdf'' decreases monotonically and is convex.
	:• <math>\lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}\,\!</math> . <math>\lambda (t)\,\!</math> is constant.		:• <math>\lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}\,\!</math> . <math>\lambda (t)\,\!</math> is constant.
	:• The mode does not exist.		:• The mode does not exist.
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	:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>		:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>
	:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>		:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>
	:• The ~~<math>~~pdf~~\,\!</math>~~ decreases monotonically and is convex.		:• The ''pdf'' decreases monotonically and is convex.
	:• As <math>\mu \,\!</math> increases, the ~~<math>~~pdf~~</math>~~ gets stretched out to the right and its height decreases, while maintaining its shape.		:• As <math>\mu \,\!</math> increases, the ''pdf'' gets stretched out to the right and its height decreases, while maintaining its shape.
	:• As <math>\mu \,\!</math> decreases, the ~~<math>~~pdf~~</math>~~ shifts towards the left and its height increases.		:• As <math>\mu \,\!</math> decreases, the ''pdf'' shifts towards the left and its height increases.
	:• The mode does not exist.		:• The mode does not exist.