Template:Probability Plotting Example (EXP1-1) - Revision history

Lisa Hacker: /* Probability Plotting Example for the 1-parameter Exponential Distribution */

2012-08-03T06:48:44Z

Probability Plotting Example for the 1-parameter Exponential Distribution

← Older revision		Revision as of 06:48, 3 August 2012
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	~~====~~Probability Plotting Example ~~for the 1-parameter Exponential Distribution====~~		#REDIRECT [[Probability Plotting Example]]
	~~Let's assume six identical units are reliability tested at the same application and operation~~
	~~stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.~~
	~~<br>~~
	~~The steps for determining the parameters of the exponential <math>pdf</math> representing the~~
	~~data, using probability plotting, are as follows:~~

	~~<br>~~
	~~Rank the times-to-failure in ascending order as shown next.~~

	~~<br>~~

	~~::<math>\begin{matrix}~~
	~~\text{Time-to-} & \text{Failure Order Number} \\~~
	~~\text{failure, hr} & \text{out of a Sample Size of 6} \\~~
	~~\text{96} & \text{1} \\~~
	~~\text{257} & \text{2} \\~~
	~~\text{498} & \text{3} \\~~
	~~\text{763} & \text{4} \\~~
	~~\text{1,051} & \text{5} \\~~
	~~\text{1,744} & \text{6} \\~~
	~~\end{matrix}</math>~~

	~~<br>~~

	~~Obtain their median rank plotting positions.~~
	~~<br>~~

	~~Median rank positions are used instead of other ranking methods because median ranks are at a~~
	~~specific confidence level (50%).~~

	~~<br>~~
	~~The times-to-failure, with their corresponding median ranks, are shown next:~~

	~~<br>~~
	~~::<math>\begin{matrix}~~
	~~\text{Time-to-} & \text{Median} \\~~
	~~\text{failure, hr} & \text{Rank, }% \\~~
	~~\text{96} & \text{10}\text{.91} \\~~
	~~\text{257} & \text{26}\text{.44} \\~~
	~~\text{498} & \text{42}\text{.14} \\~~
	~~\text{763} & \text{57}\text{.86} \\~~
	~~\text{1,051} & \text{73}\text{.56} \\~~
	~~\text{1,744} & \text{89}\text{.10} \\~~
	~~\end{matrix}</math>~~

	~~<br>~~

	~~On an exponential probability paper, plot the times on the x-axis and their corresponding~~
	~~rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The~~
	~~paper is simply a log-linear paper.~~

	~~<br>~~
	~~[[File:ALTA4.1.gif\|center]~~]
	~~<br>~~
	~~<center>Fig. 4: Sample exponential probability paper.</center>~~
	~~<br>~~

	~~Draw the best possible straight line that goes through the <math>t=0</math> and <math>~~
	~~(t)=100%</math> point and through the plotted points (as shown in Fig. 5).~~

	~~<br>~~

	~~At the <math>Q(t)=63.2%</math> or <math>R(t)=36.8%</math> 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 mean. For this case, <math>\widehat{\mu }=833</math> hr which means that <math>\lambda =\tfrac{1}{\mu }=0.0012</math> (This is always at 63.2% since <math>(T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%).</math>
	~~<br>~~
	~~[[File:ALTA4.2.gif\|center~~]]
	~~<br>~~
	~~<center>Fig. 5: Probability plot for Example 1.</center>~~
	~~<br>~~

	~~<br>~~
	~~Now any reliability value for any mission time <math>t</math> can be obtained. For example, the~~
	~~reliability for a mission of 15 hr, or any other time, can now be obtained either from the plot or analytically (i.e. using the equations given in Section <math>5.1.1</math> ).~~

	~~<br>~~
	~~To obtain the value from the plot, draw a vertical line from the abscissa, at <math>t=15</math>~~
	~~hr, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read~~
	~~<math>R(t)</math> . In this case, <math>R(t=15)=98.15%</math> . This can also be obtained~~
	~~analytically, from the exponential reliability function.~~

Lisa Hacker: /* Probability Plotting Example for the 1-parameter Exponential Distribution */

2012-07-20T12:35:15Z

Probability Plotting Example for the 1-parameter Exponential Distribution

← Older revision		Revision as of 12:35, 20 July 2012
Line 49:		Line 49:
	On an exponential probability paper, plot the times on the x-axis and their corresponding		On an exponential probability paper, plot the times on the x-axis and their corresponding
	rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The		rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The
	paper is simply a log-linear paper. ~~(The solution is given in Fig. 2.)~~		paper is simply a log-linear paper.

	<br>		<br>
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	<center>Fig. 4: Sample exponential probability paper.</center>		<center>Fig. 4: Sample exponential probability paper.</center>
	<br>		<br>

	Draw the best possible straight line that goes through the <math>t=0</math> and <math>		Draw the best possible straight line that goes through the <math>t=0</math> and <math>
	(t)=100%</math> point and through the plotted points (as shown in Fig. 5).		(t)=100%</math> point and through the plotted points (as shown in Fig. 5).

	<br>		<br>

	At the <math>Q(t)=63.2%</math> or <math>R(t)=36.8%</math> ordinate point, draw a		At the <math>Q(t)=63.2%</math> or <math>R(t)=36.8%</math> 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 mean. For this case, <math>\widehat{\mu }=833</math> hr which means that <math>\lambda =\tfrac{1}{\mu }=0.0012</math> (This is always at 63.2% since <math>(T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%).</math>		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 mean. For this case, <math>\widehat{\mu }=833</math> hr which means that <math>\lambda =\tfrac{1}{\mu }=0.0012</math> (This is always at 63.2% since <math>(T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%).</math>

Lisa Hacker: /* Probability Plotting Example for the 1-parameter Exponential Distribution */

2012-07-20T12:34:12Z

Probability Plotting Example for the 1-parameter Exponential Distribution

← Older revision		Revision as of 12:34, 20 July 2012
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	<br>		<br>

	Obtain their median rank plotting positions.		Obtain their median rank plotting positions.
	<br>		<br>

	Median rank positions are used instead of other ranking methods because median ranks are at a		Median rank positions are used instead of other ranking methods because median ranks are at a
	specific confidence level (50%).		specific confidence level (50%).

	<br>		<br>
	The times-to-failure, with their corresponding median ranks, are shown next:		The times-to-failure, with their corresponding median ranks, are shown next:
Line 43:		Line 46:

	<br>		<br>

	On an exponential probability paper, plot the times on the x-axis and their corresponding		On an exponential probability paper, plot the times on the x-axis and their corresponding
	rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The		rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The
	paper is simply a log-linear paper. (The solution is given in Fig. 2.)		paper is simply a log-linear paper. (The solution is given in Fig. 2.)

	<br>		<br>
	[[File:ALTA4.1.gif\|center]]		[[File:ALTA4.1.gif\|center]]

Lisa Hacker: /* Probability Plotting Example for the 1-parameter Exponential Distribution */

2012-07-20T12:33:42Z

Probability Plotting Example for the 1-parameter Exponential Distribution

← Older revision		Revision as of 12:33, 20 July 2012
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	The steps for determining the parameters of the exponential <math>pdf</math> representing the		The steps for determining the parameters of the exponential <math>pdf</math> representing the
	data, using probability plotting, are as follows:		data, using probability plotting, are as follows:

	<br>		<br>
	Rank the times-to-failure in ascending order as shown next.		Rank the times-to-failure in ascending order as shown next.

	<br>		<br>

Lisa Hacker: /* Probability Plotting Example for the 1-parameter Exponential Distribution */

2012-07-20T12:33:25Z

Probability Plotting Example for the 1-parameter Exponential Distribution

Lisa Hacker: /* Probability Plotting Example for the 1-parameter Exponential distribution */

2012-07-20T12:32:29Z

Probability Plotting Example for the 1-parameter Exponential distribution

← Older revision		Revision as of 12:32, 20 July 2012
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	====Probability Plotting Example for the 1-parameter Exponential ~~distribution~~====		====Probability Plotting Example for the 1-parameter Exponential Distribution====
	Let's assume six identical units are reliability tested at the same application and operation		Let's assume six identical units are reliability tested at the same application and operation
	stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.		stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.

Lisa Hacker at 10:43, 20 July 2012

2012-07-20T10:43:01Z

← Older revision		Revision as of 10:43, 20 July 2012
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	<math>R(t)</math> . In this case, <math>R(t=15)=98.15%</math> . This can also be obtained		<math>R(t)</math> . In this case, <math>R(t=15)=98.15%</math> . This can also be obtained
	analytically, from the exponential reliability function.		analytically, from the exponential reliability function.




	~~[[Category:Examples]]~~

Kate Racaza at 09:00, 5 July 2012

2012-07-05T09:00:22Z

← Older revision		Revision as of 09:00, 5 July 2012
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	====Probability Plotting Example for the 1-parameter Exponential distribution====		====Probability Plotting Example for the 1-parameter Exponential distribution====
	~~<br>~~

	Let's assume six identical units are reliability tested at the same application and operation		Let's assume six identical units are reliability tested at the same application and operation
	stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.		stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.

Pantelis at 12:55, 29 October 2011

2011-10-29T12:55:17Z

← Older revision		Revision as of 12:55, 29 October 2011
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	analytically, from the exponential reliability function.		analytically, from the exponential reliability function.

	~~<br>~~


	~~{{Template:MLE Example (EXP1-2)}}~~


	[[Category:Examples]]		[[Category:Examples]]

Pantelis at 12:52, 29 October 2011

2011-10-29T12:52:07Z

@@ Line 74: / Line 74: @@
 <br>
-<br>
+{{Template:MLE Example (EXP1-2)}}
 [[Category:Examples]]

← Older revision		Revision as of 12:33, 20 July 2012
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	data, using probability plotting, are as follows:		data, using probability plotting, are as follows:
	<br>		<br>
	~~::•~~ Rank the times-to-failure in ascending order as shown next.		Rank the times-to-failure in ascending order as shown next.
	<br>		<br>

Line 21:		Line 21:

	<br>		<br>
	~~::•~~ Obtain their median rank plotting positions.		Obtain their median rank plotting positions.
	<br>		<br>
	Median rank positions are used instead of other ranking methods because median ranks are at a		Median rank positions are used instead of other ranking methods because median ranks are at a
	specific confidence level (50%).		specific confidence level (50%).
	<br>		<br>
	~~::•~~ The times-to-failure, with their corresponding median ranks, are shown next:		The times-to-failure, with their corresponding median ranks, are shown next:

	<br>		<br>
Line 41:		Line 41:

	<br>		<br>
	~~::•~~ On an exponential probability paper, plot the times on the x-axis and their corresponding		On an exponential probability paper, plot the times on the x-axis and their corresponding
	rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The		rank value on the y-axis. Fig. 4 displays an example of an exponential probability paper. The
	paper is simply a log-linear paper. (The solution is given in Fig. 2.)		paper is simply a log-linear paper. (The solution is given in Fig. 2.)
Line 49:		Line 49:
	<center>Fig. 4: Sample exponential probability paper.</center>		<center>Fig. 4: Sample exponential probability paper.</center>
	<br>		<br>
	~~::•~~ Draw the best possible straight line that goes through the <math>t=0</math> and <math>		Draw the best possible straight line that goes through the <math>t=0</math> and <math>
	(t)=100%</math> point and through the plotted points (as shown in Fig. 5).		(t)=100%</math> point and through the plotted points (as shown in Fig. 5).
	<br>		<br>
	~~::•~~ At the <math>Q(t)=63.2%</math> or <math>R(t)=36.8%</math> ordinate point, draw a		At the <math>Q(t)=63.2%</math> or <math>R(t)=36.8%</math> 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 mean. For this case, <math>\widehat{\mu }=833</math> hr which means that <math>\lambda =\tfrac{1}{\mu }=0.0012</math> . (This is always at 63.2% since <math>(T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%).</math>		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 mean. For this case, <math>\widehat{\mu }=833</math> hr which means that <math>\lambda =\tfrac{1}{\mu }=0.0012</math> (This is always at 63.2% since <math>(T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%).</math>
	<br>		<br>
	[[File:ALTA4.2.gif\|center]]		[[File:ALTA4.2.gif\|center]]