Template:Background of weibull distribution - Revision history

Richard House: Redirected page to The Mixed Weibull Distribution

2012-08-15T00:33:28Z

Redirected page to The Mixed Weibull Distribution

← Older revision		Revision as of 00:33, 15 August 2012
Line 1:		Line 1:
	~~===Statistical Background===~~		#REDIRECT [[The_Mixed_Weibull_Distribution]]
	Consider a life test of identical components. The components were placed in a test at age <math>t=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

	If <math>N</math> identical components from a mixed population undertake a mission of <math>t</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:

	~~::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(t)}{N}</math>~~

	~~Then:~~

	~~::<math>\begin{align}~~
	~~{{N}_{1,2,...,{{n}_{S}}}}(t)= & N~~[~~{{R}_{1,2,...,n}}(t)] \\~~
	\\
	~~{{N}_{{{1}_{S}}}}(t)=& {{N}_{1}}{{R}_{1}}(t);{{N}_{{{2}_{S}}}}(t)={{N}_{2}}{{R}_{2}}(t) \\~~
	~~{{N}_{{{3}_{S}}}}(t)=& {{N}_{3}}{{R}_{3}}(t);...;{{N}_{{{n}_{S}}}}={{N}_{n}}{{R}_{n}}(t)~~
	~~\end{align}</math>~~

	~~The total number surviving by age <math>t</math> in the mixed population is the sum of the number surviving in all subpopulations or:~~


	~~::<math>{{N}_{1,2,...,{{n}_{S}}}}(t)={{N}_{{{1}_{S}}}}(t)+{{N}_{{{2}_{S}}}}(t)+{{N}_{{{3}_{S}}}}(t)+\cdots +{{N}_{{{n}_{S}}}}(t)</math>~~


	~~Substituting into the reliability equation yields:~~


	~~::<math>{{R}_{1,2,...,n}}(t)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(t)+{{N}_{2}}{{R}_{2}}(t)+{{N}_{3}}{{R}_{3}}(t)+\cdots +{{N}_{n}}{{R}_{n}}(t)]</math>~~


	~~or:~~

	~~::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>~~


	This expression can also be derived by applying Bayes' theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(t)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:

	~~::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>~~

	~~This may be written mathematically as:~~

	~~::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>~~

	~~Other functions of reliability engineering interest are found by applying the fundamentals to the above reliability equation. For example, the probability density function can be found from:~~

	~~::<math>\begin{align}~~
	~~{{f}_{1,2,...,n}}(t)= & -\frac{d}{dT}[{{R}_{1,2,...,n}}(t)] \\~~
	~~{{f}_{1,2,...,n}}(t)= & \frac{{{N}_{1}}}{N}\left( -\frac{d}{dT}[{{R}_{1}}(t)] \right)+\frac{{{N}_{2}}}{N}\left( -\frac{d}{dT}[{{R}_{2}}(t)] \right) \\~~
	~~& +\ \ \frac{{{N}_{3}}}{N}\left( -\frac{d}{dT}~~[~~{{R}_{3}}(t)~~] ~~\right)+\cdots +\frac{{{N}_{n}}}{N}\left( -\frac{d}{dT}[{{R}_{n}}(t)~~] ~~\right) \\~~
	~~{{f}_{1,2,...,n}}(t)= & \frac{{{N}_{1}}}{N}{{f}_{1}}(t)+\frac{{{N}_{2}}}{N}{{f}_{2}}(t) \\~~
	~~& +\ \ \frac{{{N}_{3}}}{N}{{f}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{f}_{n}}(t)~~
	~~\end{align}</math>~~

	~~Also, the failure rate function of a population is given by:~~

	~~::<math>\begin{align}~~
	~~{{\lambda }_{1,2,...,n}}(t)= & \frac{{{f}_{1,2,...,n}}(t)}{{{R}_{1,2,...,n}}(t)}, \\~~
	{{\lambda }_{1,2,...,n}}(t)= & \frac{\tfrac{{{N}_{1}}}{N}{{f}_{1}}(t)+\tfrac{{{N}_{2}}}{N}{{f}_{2}}(t)+\tfrac{{{N}_{3}}}{N}{{f}_{3}}(t)+\cdots +\tfrac{{{N}_{n}}}{N}{{f}_{n}}(t)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(t)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(t)+\tfrac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(t)}.
	~~\end{align}</math>~~


	The conditional reliability for a new mission of duration <math>t</math> , starting this mission at age <math>T</math> , or after having already operated a total of <math>T</math> hours, is given by:

	~~::<math>\begin{align}~~
	~~{{R}_{1,2,...,n}}(T,t)= & \frac{{{R}_{1,2,...,n}}(T+t)}{{{R}_{1,2,...,n}}(T)} \\~~
	{{R}_{1,2,...,n}}(T,t)= & \frac{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T+t)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T+t)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T+t)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T)}
	~~\end{align}</math>~~

	~~{{mixed weibull equations}}~~

Harry Guo: /* Statistic Background */

2012-02-14T22:12:39Z

Statistic Background

← Older revision		Revision as of 22:12, 14 February 2012
Line 1:		Line 1:
	===~~Statistic~~ Background===		===Statistical Background===
	Consider a life test of identical components. The components were placed in a test at age <math>t=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.		Consider a life test of identical components. The components were placed in a test at age <math>t=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

Harry Guo: /* Statistic Background */

2012-02-14T18:00:02Z

Statistic Background

← Older revision		Revision as of 18:00, 14 February 2012
Line 1:		Line 1:
	===Statistic Background===		===Statistic Background===
	Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.		Consider a life test of identical components. The components were placed in a test at age <math>t=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

	If <math>N</math> identical components from a mixed population undertake a mission of <math>t</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:		If <math>N</math> identical components from a mixed population undertake a mission of <math>t</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:
Line 32:		Line 32:


	This expression can also be derived by applying Bayes' theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(T)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:		This expression can also be derived by applying Bayes' theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(t)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:

	::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>		::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>
Line 38:		Line 38:
	This may be written mathematically as:		This may be written mathematically as:

	<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>

	Other functions of reliability engineering interest are found by applying the fundamentals to the above reliability equation. For example, the probability density function can be found from:		Other functions of reliability engineering interest are found by applying the fundamentals to the above reliability equation. For example, the probability density function can be found from:

Harry Guo: /* Statistic Background */

2012-02-14T17:57:30Z

Statistic Background

← Older revision		Revision as of 17:57, 14 February 2012
Line 16:		Line 16:

	The total number surviving by age <math>t</math> in the mixed population is the sum of the number surviving in all subpopulations or:		The total number surviving by age <math>t</math> in the mixed population is the sum of the number surviving in all subpopulations or:


	::<math>{{N}_{1,2,...,{{n}_{S}}}}(t)={{N}_{{{1}_{S}}}}(t)+{{N}_{{{2}_{S}}}}(t)+{{N}_{{{3}_{S}}}}(t)+\cdots +{{N}_{{{n}_{S}}}}(t)</math>		::<math>{{N}_{1,2,...,{{n}_{S}}}}(t)={{N}_{{{1}_{S}}}}(t)+{{N}_{{{2}_{S}}}}(t)+{{N}_{{{3}_{S}}}}(t)+\cdots +{{N}_{{{n}_{S}}}}(t)</math>
Line 21:		Line 22:

	Substituting into the reliability equation yields:		Substituting into the reliability equation yields:


	::<math>{{R}_{1,2,...,n}}(t)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(t)+{{N}_{2}}{{R}_{2}}(t)+{{N}_{3}}{{R}_{3}}(t)+\cdots +{{N}_{n}}{{R}_{n}}(t)]</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(t)+{{N}_{2}}{{R}_{2}}(t)+{{N}_{3}}{{R}_{3}}(t)+\cdots +{{N}_{n}}{{R}_{n}}(t)]</math>


	or:		or:

	::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>


	This expression can also be derived by applying Bayes' theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(T)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:		This expression can also be derived by applying Bayes' theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(T)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:

Harry Guo: /* Statistic Background */

2012-02-14T17:57:06Z

Statistic Background

← Older revision		Revision as of 17:57, 14 February 2012
Line 1:		Line 1:
	===Statistic Background===		===Statistic Background===
	Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>T=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>T=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.		Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

	If <math>N</math> identical components from a mixed population undertake a mission of <math>T</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:		If <math>N</math> identical components from a mixed population undertake a mission of <math>t</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:

	::<math>{{R}_{1,2,...,n}}(T)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(T)}{N}</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(t)}{N}</math>

	:Then:		Then:

	::<math>\begin{align}		::<math>\begin{align}
	{{N}_{1,2,...,{{n}_{S}}}}(T)= & N[{{R}_{1,2,...,n}}(T)] \\		{{N}_{1,2,...,{{n}_{S}}}}(t)= & N[{{R}_{1,2,...,n}}(t)] \\
	\\		\\
	{{N}_{{{1}_{S}}}}(T)=& {{N}_{1}}{{R}_{1}}(T);{{N}_{{{2}_{S}}}}(T)={{N}_{2}}{{R}_{2}}(T) \\		{{N}_{{{1}_{S}}}}(t)=& {{N}_{1}}{{R}_{1}}(t);{{N}_{{{2}_{S}}}}(t)={{N}_{2}}{{R}_{2}}(t) \\
	{{N}_{{{3}_{S}}}}(T)=& {{N}_{3}}{{R}_{3}}(T);...;{{N}_{{{n}_{S}}}}={{N}_{n}}{{R}_{n}}(T)		{{N}_{{{3}_{S}}}}(t)=& {{N}_{3}}{{R}_{3}}(t);...;{{N}_{{{n}_{S}}}}={{N}_{n}}{{R}_{n}}(t)
	\end{align}</math>		\end{align}</math>

	The total number surviving by age <math>T</math> in the mixed population is the sum of the number surviving in all subpopulations or:		The total number surviving by age <math>t</math> in the mixed population is the sum of the number surviving in all subpopulations or:

	::<math>{{N}_{1,2,...,{{n}_{S}}}}(T)={{N}_{{{1}_{S}}}}(T)+{{N}_{{{2}_{S}}}}(T)+{{N}_{{{3}_{S}}}}(T)+\cdots +{{N}_{{{n}_{S}}}}(T)</math>		::<math>{{N}_{1,2,...,{{n}_{S}}}}(t)={{N}_{{{1}_{S}}}}(t)+{{N}_{{{2}_{S}}}}(t)+{{N}_{{{3}_{S}}}}(t)+\cdots +{{N}_{{{n}_{S}}}}(t)</math>


	Substituting into ~~Eqn. (rel)~~ yields:		Substituting into the reliability equation yields:

	::<math>{{R}_{1,2,...,n}}(T)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(T)+{{N}_{2}}{{R}_{2}}(T)+{{N}_{3}}{{R}_{3}}(T)+\cdots +{{N}_{n}}{{R}_{n}}(T)]</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(t)+{{N}_{2}}{{R}_{2}}(t)+{{N}_{3}}{{R}_{3}}(t)+\cdots +{{N}_{n}}{{R}_{n}}(t)]</math>

	:or:		or:

	::<math>{{R}_{1,2,...,n}}(T)=\frac{{{N}_{1}}}{N}{{R}_{1}}(T)+\frac{{{N}_{2}}}{N}{{R}_{2}}(T)+\frac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(T)</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>

	This expression can also be derived by applying Bayes' theorem [20], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(T)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(T)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(T)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(T)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:		This expression can also be derived by applying Bayes' theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(T)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:

	::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>		::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>
Line 34:		Line 34:
	This may be written mathematically as:		This may be written mathematically as:

	<math>{{R}_{1,2,...,n}}(T)=\frac{{{N}_{1}}}{N}{{R}_{1}}(T)+\frac{{{N}_{2}}}{N}{{R}_{2}}(T)+\frac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(T)</math>		<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>

	Other functions of reliability engineering interest are found by applying the fundamentals to ~~Eqn~~. ~~(rel1).~~		Other functions of reliability engineering interest are found by applying the fundamentals to the above reliability equation. For example, the probability density function can be found from:

	For example, the probability density function can be found from:

	::<math>\begin{align}		::<math>\begin{align}
	{{f}_{1,2,...,n}}(T)= & -\frac{d}{dT}[{{R}_{1,2,...,n}}(T)] \\		{{f}_{1,2,...,n}}(t)= & -\frac{d}{dT}[{{R}_{1,2,...,n}}(t)] \\
	{{f}_{1,2,...,n}}(T)= & \frac{{{N}_{1}}}{N}\left( -\frac{d}{dT}[{{R}_{1}}(T)] \right)+\frac{{{N}_{2}}}{N}\left( -\frac{d}{dT}[{{R}_{2}}(T)] \right) \\		{{f}_{1,2,...,n}}(t)= & \frac{{{N}_{1}}}{N}\left( -\frac{d}{dT}[{{R}_{1}}(t)] \right)+\frac{{{N}_{2}}}{N}\left( -\frac{d}{dT}[{{R}_{2}}(t)] \right) \\
	& +\ \ \frac{{{N}_{3}}}{N}\left( -\frac{d}{dT}[{{R}_{3}}(T)] \right)+\cdots +\frac{{{N}_{n}}}{N}\left( -\frac{d}{dT}[{{R}_{n}}(T)] \right) \\		& +\ \ \frac{{{N}_{3}}}{N}\left( -\frac{d}{dT}[{{R}_{3}}(t)] \right)+\cdots +\frac{{{N}_{n}}}{N}\left( -\frac{d}{dT}[{{R}_{n}}(t)] \right) \\
	{{f}_{1,2,...,n}}(T)= & \frac{{{N}_{1}}}{N}{{f}_{1}}(T)+\frac{{{N}_{2}}}{N}{{f}_{2}}(T) \\		{{f}_{1,2,...,n}}(t)= & \frac{{{N}_{1}}}{N}{{f}_{1}}(t)+\frac{{{N}_{2}}}{N}{{f}_{2}}(t) \\
	& +\ \ \frac{{{N}_{3}}}{N}{{f}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{f}_{n}}(T)		& +\ \ \frac{{{N}_{3}}}{N}{{f}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{f}_{n}}(t)
	\end{align}</math>		\end{align}</math>

Line 51:		Line 49:

	::<math>\begin{align}		::<math>\begin{align}
	{{\lambda }_{1,2,...,n}}(T)= & \frac{{{f}_{1,2,...,n}}(T)}{{{R}_{1,2,...,n}}(T)}, \\		{{\lambda }_{1,2,...,n}}(t)= & \frac{{{f}_{1,2,...,n}}(t)}{{{R}_{1,2,...,n}}(t)}, \\
	{{\lambda }_{1,2,...,n}}(T)= & \frac{\tfrac{{{N}_{1}}}{N}{{f}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{f}_{2}}(T)+\tfrac{{{N}_{3}}}{N}{{f}_{3}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{f}_{n}}(T)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T)+\tfrac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T)}.		{{\lambda }_{1,2,...,n}}(t)= & \frac{\tfrac{{{N}_{1}}}{N}{{f}_{1}}(t)+\tfrac{{{N}_{2}}}{N}{{f}_{2}}(t)+\tfrac{{{N}_{3}}}{N}{{f}_{3}}(t)+\cdots +\tfrac{{{N}_{n}}}{N}{{f}_{n}}(t)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(t)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(t)+\tfrac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(t)}.
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Background */

2012-02-14T17:37:37Z

Background

← Older revision		Revision as of 17:37, 14 February 2012
Line 1:		Line 1:
	===Background===		===Statistic Background===
	Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>T=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>T=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.		Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>T=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>T=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

Nicolette Young: /* The Mixed Weibull Equations */

2012-01-24T16:32:37Z

The Mixed Weibull Equations

← Older revision		Revision as of 16:32, 24 January 2012
Line 63:		Line 63:
	\end{align}</math>		\end{align}</math>

	~~====The Mixed Weibull Equations====~~		{{mixed weibull equations}}
	~~Depending on the number of subpopulations chosen, Weibull++ uses the following equations for the reliability and probability density functions:~~


	~~::<math>~~{{~~R}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{T}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}}</math>~~

	~~:and:~~

	::<math>{{f}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{T}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{T}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}}~~</math>~~

	where <math>S=2</math> , <math>S=3</math> , and <math>S=4</math> for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.

Nicolette Young: Created page with '===Background=== Consider a life test of identical components. The components were placed in a test at age $T=0$ and were tested to failure, with their times-to-fai…'

2012-01-04T18:40:19Z

Created page with '===Background=== Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-fai…'

New page

===Background===
Consider a life test of identical components. The components were placed in a test at age <math>T=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>T=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>T=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

If <math>N</math> identical components from a mixed population undertake a mission of <math>T</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:

::<math>{{R}_{1,2,...,n}}(T)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(T)}{N}</math>

:Then:

::<math>\begin{align}
{{N}_{1,2,...,{{n}_{S}}}}(T)= & N[{{R}_{1,2,...,n}}(T)] \\
\\
{{N}_{{{1}_{S}}}}(T)=& {{N}_{1}}{{R}_{1}}(T);{{N}_{{{2}_{S}}}}(T)={{N}_{2}}{{R}_{2}}(T) \\
{{N}_{{{3}_{S}}}}(T)=& {{N}_{3}}{{R}_{3}}(T);...;{{N}_{{{n}_{S}}}}={{N}_{n}}{{R}_{n}}(T)
\end{align}</math>

The total number surviving by age <math>T</math> in the mixed population is the sum of the number surviving in all subpopulations or:

::<math>{{N}_{1,2,...,{{n}_{S}}}}(T)={{N}_{{{1}_{S}}}}(T)+{{N}_{{{2}_{S}}}}(T)+{{N}_{{{3}_{S}}}}(T)+\cdots +{{N}_{{{n}_{S}}}}(T)</math>

Substituting into Eqn. (rel) yields:

::<math>{{R}_{1,2,...,n}}(T)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(T)+{{N}_{2}}{{R}_{2}}(T)+{{N}_{3}}{{R}_{3}}(T)+\cdots +{{N}_{n}}{{R}_{n}}(T)]</math>

:or:

::<math>{{R}_{1,2,...,n}}(T)=\frac{{{N}_{1}}}{N}{{R}_{1}}(T)+\frac{{{N}_{2}}}{N}{{R}_{2}}(T)+\frac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(T)</math>

This expression can also be derived by applying Bayes' theorem [20], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(T)</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(T)</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(T)</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(T)</math> , given that the component is from subpopulation <math>n</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:

::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>

This may be written mathematically as:

<math>{{R}_{1,2,...,n}}(T)=\frac{{{N}_{1}}}{N}{{R}_{1}}(T)+\frac{{{N}_{2}}}{N}{{R}_{2}}(T)+\frac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(T)</math>

Other functions of reliability engineering interest are found by applying the fundamentals to Eqn. (rel1).

For example, the probability density function can be found from:

::<math>\begin{align}
{{f}_{1,2,...,n}}(T)= & -\frac{d}{dT}[{{R}_{1,2,...,n}}(T)] \\
{{f}_{1,2,...,n}}(T)= & \frac{{{N}_{1}}}{N}\left( -\frac{d}{dT}[{{R}_{1}}(T)] \right)+\frac{{{N}_{2}}}{N}\left( -\frac{d}{dT}[{{R}_{2}}(T)] \right) \\
& +\ \ \frac{{{N}_{3}}}{N}\left( -\frac{d}{dT}[{{R}_{3}}(T)] \right)+\cdots +\frac{{{N}_{n}}}{N}\left( -\frac{d}{dT}[{{R}_{n}}(T)] \right) \\
{{f}_{1,2,...,n}}(T)= & \frac{{{N}_{1}}}{N}{{f}_{1}}(T)+\frac{{{N}_{2}}}{N}{{f}_{2}}(T) \\
& +\ \ \frac{{{N}_{3}}}{N}{{f}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{f}_{n}}(T)
\end{align}</math>

Also, the failure rate function of a population is given by:

::<math>\begin{align}
{{\lambda }_{1,2,...,n}}(T)= & \frac{{{f}_{1,2,...,n}}(T)}{{{R}_{1,2,...,n}}(T)}, \\
{{\lambda }_{1,2,...,n}}(T)= & \frac{\tfrac{{{N}_{1}}}{N}{{f}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{f}_{2}}(T)+\tfrac{{{N}_{3}}}{N}{{f}_{3}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{f}_{n}}(T)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T)+\tfrac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T)}.
\end{align}</math>

The conditional reliability for a new mission of duration <math>t</math> , starting this mission at age <math>T</math> , or after having already operated a total of <math>T</math> hours, is given by:

::<math>\begin{align}
{{R}_{1,2,...,n}}(T,t)= & \frac{{{R}_{1,2,...,n}}(T+t)}{{{R}_{1,2,...,n}}(T)} \\
{{R}_{1,2,...,n}}(T,t)= & \frac{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T+t)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T+t)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T+t)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T)}
\end{align}</math>

====The Mixed Weibull Equations====
Depending on the number of subpopulations chosen, Weibull++ uses the following equations for the reliability and probability density functions:

::<math>{{R}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{T}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}}</math>

:and:

::<math>{{f}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{T}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{T}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}}</math>

where <math>S=2</math> , <math>S=3</math> , and <math>S=4</math> for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.

Template:Background of weibull distribution - Revision history

Richard House: Redirected page to The Mixed Weibull Distribution

Harry Guo: /* Statistic Background */

Harry Guo: /* Statistic Background */

Harry Guo: /* Statistic Background */

Harry Guo: /* Statistic Background */

Harry Guo: /* Background */

Nicolette Young: /* The Mixed Weibull Equations */

Nicolette Young: Created page with '===Background=== Consider a life test of identical components. The components were placed in a test at age T=0 and were tested to failure, with their times-to-fai…'

Nicolette Young: Created page with '===Background=== Consider a life test of identical components. The components were placed in a test at age $T=0$ and were tested to failure, with their times-to-fai…'