Exponential Distribution Functions - Revision history

Lisa Hacker at 21:41, 18 September 2023

2023-09-18T21:41:02Z

← Older revision		Revision as of 21:41, 18 September 2023
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	<noinclude>{{Navigation box}}		<noinclude>{{Navigation box}}
	''This article also appears in the [~~[The_Exponential_Distribution\|~~Life ~~Data Analysis Reference]~~] and [~~[Distributions_Used_in_Accelerated_Testing\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] ~~books~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference] and [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference].'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean, <math>\overline{T},\,\!</math> or mean time to failure (MTTF) is given by:		The mean, <math>\overline{T},\,\!</math> or mean time to failure (MTTF) is given by:

Richard House at 18:05, 24 September 2012

2012-09-24T18:05:49Z

← Older revision		Revision as of 18:05, 24 September 2012
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	''This article also appears in the [[The_Exponential_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Exponential_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:		The mean, <math>\overline{T},\,\!</math> or mean time to failure (MTTF) is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	= & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\		= & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\
	= & \gamma +\frac{1}{\lambda }=m		= & \gamma +\frac{1}{\lambda }=m
	\end{align}</math>		\end{align}\,\!</math>

	Note that when <math>\gamma =0\,\!</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.		Note that when <math>\gamma =0\,\!</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.

	===The Median===		===The Median===
	The median, <math> \breve{T}, </math> is:		The median, <math> \breve{T}, \,\!</math> is:

	::<math> \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 </math>		::<math> \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 \,\!</math>

	===The Mode===		===The Mode===
	The mode, <math>\tilde{T},</math> is:		The mode, <math>\tilde{T},\,\!</math> is:

	::<math>\tilde{T}=\gamma </math>		::<math>\tilde{T}=\gamma \,\!</math>

	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation, <math>{\sigma }_{T}\,\!</math>, is:		The standard deviation, <math>{\sigma }_{T}\,\!</math>, is:

	::<math>{\sigma}_{T}=\frac{1}{\lambda }=m</math>		::<math>{\sigma}_{T}=\frac{1}{\lambda }=m\,\!</math>

	===The Exponential Reliability Function===		===The Exponential Reliability Function===
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	::<math>\begin{align}		::<math>\begin{align}
	F(t)=Q(t)=1-{{e}^{-\lambda (t-\gamma )}}		F(t)=Q(t)=1-{{e}^{-\lambda (t-\gamma )}}
	\end{align}</math>		\end{align}\,\!</math>

	Recalling that the reliability function of a distribution is simply one minus the ''cdf'', the reliability function of the 2-parameter exponential distribution is given by:		Recalling that the reliability function of a distribution is simply one minus the ''cdf'', the reliability function of the 2-parameter exponential distribution is given by:

	::<math>R(t)=1-Q(t)=1-\int_{0}^{t-\gamma }f(x)dx</math>		::<math>R(t)=1-Q(t)=1-\int_{0}^{t-\gamma }f(x)dx\,\!</math>


	::<math>R(t)=1-\int_{0}^{t-\gamma }\lambda {{e}^{-\lambda x}}dx={{e}^{-\lambda (t-\gamma )}}</math>		::<math>R(t)=1-\int_{0}^{t-\gamma }\lambda {{e}^{-\lambda x}}dx={{e}^{-\lambda (t-\gamma )}}\,\!</math>

	The 1-parameter exponential reliability function is given by:		The 1-parameter exponential reliability function is given by:

	::<math>R(t)={{e}^{-\lambda t}}={{e}^{-\tfrac{t}{m}}}</math>		::<math>R(t)={{e}^{-\lambda t}}={{e}^{-\tfrac{t}{m}}}\,\!</math>

	===The Exponential Conditional Reliability Function===		===The Exponential Conditional Reliability Function===
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	The exponential conditional reliability equation gives the reliability for a mission of <math>t\,\!</math> duration, having already successfully accumulated <math>T\,\!</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:		The exponential conditional reliability equation gives the reliability for a mission of <math>t\,\!</math> duration, having already successfully accumulated <math>T\,\!</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:

	::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}\,\!</math>

	which says that the reliability for a mission of <math>t\,\!</math> duration undertaken after the component or equipment has already accumulated <math>T\,\!</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ''memoryless property''.		which says that the reliability for a mission of <math>t\,\!</math> duration undertaken after the component or equipment has already accumulated <math>T\,\!</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ''memoryless property''.
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	The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}\,\!</math>, for the 1-parameter exponential distribution is:		The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}\,\!</math>, for the 1-parameter exponential distribution is:

	::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>		::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}\,\!</math>

	::<math>\begin{align}		::<math>\begin{align}
	\ln[R({{t}_{R}})]=-\lambda({{t}_{R}}-\gamma )		\ln[R({{t}_{R}})]=-\lambda({{t}_{R}}-\gamma )
	\end{align}</math>		\end{align}\,\!</math>

	or:		or:

	::<math>{{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda }</math>		::<math>{{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda }\,\!</math>

	===The Exponential Failure Rate Function===		===The Exponential Failure Rate Function===
	The exponential failure rate function is:		The exponential failure rate function is:

	::<math>\lambda (t)=\frac{f(t)}{R(t)}=\frac{\lambda {{e}^{-\lambda (t-\gamma )}}}{{{e}^{-\lambda (t-\gamma )}}}=\lambda =\text{constant}</math>		::<math>\lambda (t)=\frac{f(t)}{R(t)}=\frac{\lambda {{e}^{-\lambda (t-\gamma )}}}{{{e}^{-\lambda (t-\gamma )}}}=\lambda =\text{constant}\,\!</math>

	Once again, note that the constant failure rate is a characteristic of the exponential distribution, and special cases of other distributions only. Most other distributions have failure rates that are functions of time.		Once again, note that the constant failure rate is a characteristic of the exponential distribution, and special cases of other distributions only. Most other distributions have failure rates that are functions of time.

Richard House: /* The Exponential Reliability Function */

2012-08-29T20:22:56Z

The Exponential Reliability Function

← Older revision		Revision as of 20:22, 29 August 2012
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	===The Exponential Reliability Function===		===The Exponential Reliability Function===
	The equation for the 2-parameter exponential cumulative density function, or ~~<math>~~cdf,~~</math>~~ is given by:		The equation for the 2-parameter exponential cumulative density function, or ''cdf'', is given by:

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

	Recalling that the reliability function of a distribution is simply one minus the ~~<math>~~cdf~~</math>~~, the reliability function of the 2-parameter exponential distribution is given by:		Recalling that the reliability function of a distribution is simply one minus the ''cdf'', the reliability function of the 2-parameter exponential distribution is given by:

	::<math>R(t)=1-Q(t)=1-\int_{0}^{t-\gamma }f(x)dx</math>		::<math>R(t)=1-Q(t)=1-\int_{0}^{t-\gamma }f(x)dx</math>

Richard House: /* The Exponential Conditional Reliability Function */

2012-08-22T00:26:19Z

The Exponential Conditional Reliability Function

← Older revision		Revision as of 00:26, 22 August 2012
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	===The Exponential Conditional Reliability Function===		===The Exponential Conditional Reliability Function===

	The exponential conditional reliability equation gives the reliability for a mission of <math>t</math> duration, having already successfully accumulated <math>T</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:		The exponential conditional reliability equation gives the reliability for a mission of <math>t\,\!</math> duration, having already successfully accumulated <math>T\,\!</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:

	::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>

	which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ''memoryless property''.		which says that the reliability for a mission of <math>t\,\!</math> duration undertaken after the component or equipment has already accumulated <math>T\,\!</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ''memoryless property''.

	===The Exponential Reliable Life Function===		===The Exponential Reliable Life Function===

Richard House: /* The Exponential Reliable Life Function */

2012-08-22T00:25:26Z

The Exponential Reliable Life Function

← Older revision		Revision as of 00:25, 22 August 2012
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	===The Exponential Reliable Life Function===		===The Exponential Reliable Life Function===

	The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math>, for the 1-parameter exponential distribution is:		The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}\,\!</math>, for the 1-parameter exponential distribution is:

	::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>		::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>

Richard House: /* The Standard Deviation */

2012-08-22T00:25:00Z

The Standard Deviation

← Older revision		Revision as of 00:25, 22 August 2012
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	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation, <math>{\sigma }_{T}</math>, is:		The standard deviation, <math>{\sigma }_{T}\,\!</math>, is:

	::<math>{\sigma}_{T}=\frac{1}{\lambda }=m</math>		::<math>{\sigma}_{T}=\frac{1}{\lambda }=m</math>

Richard House: /* The Mean or MTTF */

2012-08-22T00:24:34Z

The Mean or MTTF

← Older revision		Revision as of 00:24, 22 August 2012
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	\end{align}</math>		\end{align}</math>

	Note that when <math>\gamma =0</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.		Note that when <math>\gamma =0\,\!</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.

	===The Median===		===The Median===

Richard House at 01:57, 8 August 2012

2012-08-08T01:57:37Z

← Older revision		Revision as of 01:57, 8 August 2012
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	===The Mean or MTTF===		===The Mean or MTTF===
	The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:		The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:


	::<math>\begin{align}		::<math>\begin{align}
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	= & \gamma +\frac{1}{\lambda }=m		= & \gamma +\frac{1}{\lambda }=m
	\end{align}</math>		\end{align}</math>


	Note that when <math>\gamma =0</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.		Note that when <math>\gamma =0</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.
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	::<math> \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 </math>		::<math> \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 </math>


	===The Mode===		===The Mode===

Lisa Hacker: /* The Exponential Conditional Reliability Function */

2012-08-07T03:31:30Z

The Exponential Conditional Reliability Function

← Older revision		Revision as of 03:31, 7 August 2012
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	::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>

	which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the memoryless property.		which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ''memoryless property''.

	===The Exponential Reliable Life Function===		===The Exponential Reliable Life Function===

Lisa Hacker at 02:22, 7 August 2012

2012-08-07T02:22:39Z

← Older revision		Revision as of 02:22, 7 August 2012
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			<noinclude>{{Navigation box}}
			''This article also appears in the [[The_Exponential_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	===The Mean or MTTF===		===The Mean or MTTF===
	The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:		The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by: