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Richard House: Redirected page to Distributions Used in Accelerated Testing#The Lognormal Distribution

2012-08-16T03:23:03Z

Redirected page to Distributions Used in Accelerated Testing#The Lognormal Distribution

Richard House at 03:06, 16 August 2012

2012-08-16T03:06:51Z

← Older revision		Revision as of 03:06, 16 August 2012
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	<br>		<br>

	~~{{alta ld statistical prop func}}~~

	{{ald characteristics}}		{{ald characteristics}}

Richard House at 01:52, 6 August 2012

2012-08-06T01:52:21Z

← Older revision		Revision as of 01:52, 6 August 2012
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	<br>		<br>
	~~{{RS Copyright}}~~
	[[Category:Accelerated_Life_Data_Analysis_Reference]]		[[Category:Accelerated_Life_Data_Analysis_Reference]]

Chris Kahn: /* Example 5 */

2012-06-25T10:10:11Z

Example 5

← Older revision		Revision as of 10:10, 25 June 2012
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	:* On a lognormal probability paper, plot the times and their corresponding rank value. The next figure displays an example of a lognormal probability paper. The paper is simply a log-log paper.		:* On a lognormal probability paper, plot the times and their corresponding rank value. The next figure displays an example of a lognormal probability paper. The paper is simply a log-log paper.

	<br> [[Image:ALTA4.10.png\|center\|~~200px~~]] <br>		<br> [[Image:ALTA4.10.png\|center\|400px]] <br>

	:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.		:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.
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	::		::

	<br> [[Image:ALTA4.11.png\|center\|~~200px~~]] <br> <br>		<br> [[Image:ALTA4.11.png\|center\|400px]] <br> <br>

	:* The standard deviation, <span class="texhtml">σ<sub>''T'''</sub>,</span> can be found using the following equation:		:* The standard deviation, <span class="texhtml">σ<sub>''T'''</sub>,</span> can be found using the following equation:

Chuck Smith: /* Example 5 */

2012-04-26T16:08:52Z

Example 5

← Older revision		Revision as of 16:08, 26 April 2012
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	:* On a lognormal probability paper, plot the times and their corresponding rank value. The next figure displays an example of a lognormal probability paper. The paper is simply a log-log paper.		:* On a lognormal probability paper, plot the times and their corresponding rank value. The next figure displays an example of a lognormal probability paper. The paper is simply a log-log paper.

	<br> [[Image:ALTA4.10.png\|center\|~~400px~~]] <br>		<br> [[Image:ALTA4.10.png\|center\|200px]] <br>

	:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.		:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.
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	::		::

	<br> [[Image:ALTA4.11.png\|center\|~~400px~~]] <br> <br>		<br> [[Image:ALTA4.11.png\|center\|200px]] <br> <br>

	:* The standard deviation, <span class="texhtml">σ<sub>''T'''</sub>,</span> can be found using the following equation:		:* The standard deviation, <span class="texhtml">σ<sub>''T'''</sub>,</span> can be found using the following equation:

Nicolette Young: /* Example 5 */

2012-03-20T17:17:50Z

Example 5

← Older revision		Revision as of 17:17, 20 March 2012
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	:* On a lognormal probability paper, plot the times and their corresponding rank value. The next figure displays an example of a lognormal probability paper. The paper is simply a log-log paper.		:* On a lognormal probability paper, plot the times and their corresponding rank value. The next figure displays an example of a lognormal probability paper. The paper is simply a log-log paper.

	<br> [[Image:ALTA4.10.~~gif\|thumb~~\|center\|~~300px~~]] <br>		<br> [[Image:ALTA4.10.png\|center\|400px]] <br>

	:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.		:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.
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	::		::

	<br> [[Image:ALTA4.11.~~gif\|thumb~~\|center\|~~300px~~]] <br> <br>		<br> [[Image:ALTA4.11.png\|center\|400px]] <br> <br>

	:* The standard deviation, <span class="texhtml">σ<sub>''T'''</sub>,</span> can be found using the following equation:		:* The standard deviation, <span class="texhtml">σ<sub>''T'''</sub>,</span> can be found using the following equation:

Chris Kahn: /* Example 6 */

2012-03-07T17:43:36Z

Example 6

← Older revision		Revision as of 17:43, 7 March 2012
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	<br>		<br>
	{{RS Copyright}}		{{RS Copyright}}
	[[Category:~~Accelerated_Testing_Reference~~]]		[[Category:Accelerated_Life_Data_Analysis_Reference]]

Chris Kahn: /* Example 5 */

2012-03-07T17:36:55Z

Example 5

← Older revision		Revision as of 17:36, 7 March 2012
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	:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.		:* Draw the best possible straight line that goes through the <span class="texhtml">''t'' = 0</span> and <span class="texhtml">''R''(''t'') = 100%</span> point and through these points.
	:* At the <span class="texhtml">''Q''(''t'') = 50%</span> ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of the median. For this case, <math>\breve{T}=760</math> hr which means that <math>{{\bar{T}}^{\prime }}=\ln(\breve{T})=6.633</math>.		:* At the <span class="texhtml">''Q''(''t'') = 50%</span> ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of the median. For this case, <math>\breve{T}=760</math> hours which means that <math>{{\bar{T}}^{\prime }}=\ln(\breve{T})=6.633</math>.

	<br>		<br>
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	\end{align}</math>		\end{align}</math>

	<br> Now any reliability value for any mission time <span class="texhtml">''t''</span> can be obtained. For example, the reliability for a mission of 200 hr, or any other time, can now be obtained either from the plot or analytically. <br> To obtain the value from the plot, draw a vertical line from the abscissa, at <span class="texhtml">''t'' = 200</span> hr, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span> . In this case, <span class="texhtml">''R''(''t'' = 200) = 1 − ''Q''(''t'' = 200) = 92%</span> . This can also be obtained analytically, from the lognormal reliability function. However, standard normal tables (or the Quick Statistical Reference in ALTA) must be used.		<br> Now any reliability value for any mission time <span class="texhtml">''t''</span> can be obtained. For example, the reliability for a mission of 200 hours, or any other time, can now be obtained either from the plot or analytically. <br> To obtain the value from the plot, draw a vertical line from the abscissa, at <span class="texhtml">''t'' = 200</span> hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span> . In this case, <span class="texhtml">''R''(''t'' = 200) = 1 − ''Q''(''t'' = 200) = 92%</span> . This can also be obtained analytically, from the lognormal reliability function. However, standard normal tables (or the Quick Statistical Reference in ALTA) must be used.

	{{ald mle}}		{{ald mle}}

Chris Kahn: /* Example 6 */

2012-03-07T17:24:38Z

Example 6

← Older revision		Revision as of 17:24, 7 March 2012
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	===== Example 6 =====		===== Example 6 =====

			Using the same data as in the probability plotting example (Example 5) and assuming a lognormal distribution, estimate the parameters using the MLE method.

	'''Solution '''		'''Solution '''

Chris Kahn: /* Example 6 */

2012-03-07T17:22:01Z

Example 6

← Older revision		Revision as of 17:22, 7 March 2012
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	\end{align}</math>		\end{align}</math>

			<br>
	{{RS Copyright}}		{{RS Copyright}}
	~~<br>~~
	[[Category:Accelerated_Testing_Reference]]		[[Category:Accelerated_Testing_Reference]]