Crow Extended Confidence Bounds Example 2 - Revision history

Lisa Hacker at 21:31, 18 September 2023

2023-09-18T21:31:02Z

← Older revision		Revision as of 21:31, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow_Extended\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 22:30, 15 May 2017

2017-05-15T22:30:12Z

← Older revision		Revision as of 22:30, 15 May 2017
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	:<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\		{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{CA}}(T)}} \\
	= & 0.09339		= & 0.09339
	\end{align}\,\!</math>		\end{align}\,\!</math>

	:<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\		{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{CA}}(T)}} \\
	= & 0.17390		= & 0.17390
	\end{align}\,\!</math>		\end{align}\,\!</math>

David J. Groebel at 17:34, 30 May 2014

2014-05-30T17:34:20Z

← Older revision		Revision as of 17:34, 30 May 2014
Line 9:		Line 9:
	<br>		<br>
	For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, <math>{{\hat{\lambda }}_{D}}(T)\,\!</math>, is simply the Crow-AMSAA model applied to all A, BC, and BD data.		For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, <math>{{\hat{\lambda }}_{D}}(T)\,\!</math>, is simply the Crow-AMSAA model applied to all A, BC, and BD data.


	:<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\!</math>		:<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\!</math>


	Therefore, the demonstrated MTBF is:		Therefore, the demonstrated MTBF is:


	:<math>MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\!</math>		:<math>MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\!</math>


	Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:		Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 27:		Line 22:
	= & 0.09339		= & 0.09339
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 33:		Line 27:
	= & 0.17390		= & 0.17390
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:		The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 44:		Line 36:
	= & 10.70799		= & 10.70799
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:		The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 57:		Line 47:
	= & 10.8779		= & 10.8779
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The projected failure intensity is:		The projected failure intensity is:


	:<math>\begin{align}		:<math>\begin{align}
Line 66:		Line 54:
	&= 0.0885 \,\!		&= 0.0885 \,\!
	\end{align}</math>		\end{align}</math>


	Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:		Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:


	:<math>\begin{align}		:<math>\begin{align}
Line 75:		Line 61:
	= & 0.0681		= & 0.0681
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 81:		Line 66:
	= & 0.1152		= & 0.1152
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:		The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 92:		Line 75:
	= & 14.6926		= & 14.6926
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Crow confidence bounds for the projected failure intensity are:		The Crow confidence bounds for the projected failure intensity are:


	:<math>\begin{align}		:<math>\begin{align}
Line 103:		Line 84:
	= & 0.1166		= & 0.1166
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Crow confidence bounds for the projected MTBF at the 90% confidence level are:		The Crow confidence bounds for the projected MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 114:		Line 93:
	= & 14.8769		= & 14.8769
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The growth potential failure intensity is:		The growth potential failure intensity is:


	:<math>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\!</math>		:<math>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\!</math>


	Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:		Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:


	:<math>\begin{align}		:<math>\begin{align}
Line 131:		Line 106:
	= & 0.0919		= & 0.0919
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:		The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 142:		Line 115:
	= & 20.4855		= & 20.4855
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.		The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.

	[[Image:rga9.10.png\|center\|~~400px\|Fisher Matrix confidence bounds on MTBF.~~]]		[[Image:rga9.10.png\|center\|450px]]

	The next figure shows these bounds based on the Crow method.		The next figure shows these bounds based on the Crow method.

	[[Image:rga9.11.png\|center\|~~400px\|Crow confidence bounds on MTBF.~~]]		[[Image:rga9.11.png\|center\|450px]]

David J. Groebel at 23:19, 22 April 2014

2014-04-22T23:19:44Z

← Older revision		Revision as of 23:19, 22 April 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow_Extended\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Crow_Extended\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

Chris Kahn at 23:58, 29 January 2014

2014-01-29T23:58:12Z

← Older revision		Revision as of 23:58, 29 January 2014
Line 9:		Line 9:
	<br>		<br>
	For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, <math>{{\hat{\lambda }}_{D}}(T)\,\!</math>, is simply the Crow-AMSAA model applied to all A, BC, and BD data.		For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, <math>{{\hat{\lambda }}_{D}}(T)\,\!</math>, is simply the Crow-AMSAA model applied to all A, BC, and BD data.


	:<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\!</math>		:<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\!</math>


	Therefore, the demonstrated MTBF is:		Therefore, the demonstrated MTBF is:


	:<math>MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\!</math>		:<math>MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\!</math>


	Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:		Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 22:		Line 27:
	= & 0.09339		= & 0.09339
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 27:		Line 33:
	= & 0.17390		= & 0.17390
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:		The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 36:		Line 44:
	= & 10.70799		= & 10.70799
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:		The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 47:		Line 57:
	= & 10.8779		= & 10.8779
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The projected failure intensity is:		The projected failure intensity is:


	:<math>\begin{align}		:<math>\begin{align}
Line 54:		Line 66:
	&= 0.0885 \,\!		&= 0.0885 \,\!
	\end{align}</math>		\end{align}</math>


	Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:		Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:


	:<math>\begin{align}		:<math>\begin{align}
Line 62:		Line 76:
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\		{{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
	= & 0.1152		= & 0.1152
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:		The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 75:		Line 92:
	= & 14.6926		= & 14.6926
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Crow confidence bounds for the projected failure intensity are:		The Crow confidence bounds for the projected failure intensity are:


	:<math>\begin{align}		:<math>\begin{align}
Line 84:		Line 103:
	= & 0.1166		= & 0.1166
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Crow confidence bounds for the projected MTBF at the 90% confidence level are:		The Crow confidence bounds for the projected MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 93:		Line 114:
	= & 14.8769		= & 14.8769
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The growth potential failure intensity is:		The growth potential failure intensity is:


	:<math>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\!</math>		:<math>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\!</math>


	Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:		Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:


	:<math>\begin{align}		:<math>\begin{align}
Line 106:		Line 131:
	= & 0.0919		= & 0.0919
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:		The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:


	:<math>\begin{align}		:<math>\begin{align}
Line 115:		Line 142:
	= & 20.4855		= & 20.4855
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.		The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.

David J. Groebel at 18:12, 27 January 2014

2014-01-27T18:12:33Z

Show changes

Kate Racaza at 19:47, 20 January 2014

2014-01-20T19:47:09Z

← Older revision		Revision as of 19:47, 20 January 2014
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	</noinclude>		</noinclude>

	Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the Test-Fix-Find-Test data <noinclude>in the [[~~Crow_Extended#Example:_Test~~-Fix-Find-~~Test_Data~~\|Crow Extended Test-Fix-Find-Test Example]]</noinclude><includeonly>given above</includeonly>.		Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the Test-Fix-Find-Test data <noinclude>in the [[Test-Fix-Find-Test_Data_Example\|Crow Extended Test-Fix-Find-Test Example]]</noinclude><includeonly>given above</includeonly>.
	<br>		<br>
	<br>		<br>

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. Calculate the…'

2014-01-20T19:38:08Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> Calculate the…'

New page

<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Crow_Extended|Reliability Growth and Repairable System Analysis Reference book]]''.
</noinclude>

Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the Test-Fix-Find-Test data <noinclude>in the [[Crow_Extended#Example:_Test-Fix-Find-Test_Data|Crow Extended Test-Fix-Find-Test Example]]</noinclude><includeonly>given above</includeonly>.
<br>
<br>
'''Solution'''
<br>
For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, <math>{{\hat{\lambda }}_{D}}(T)\,\!</math>, is simply the Crow-AMSAA model applied to all A, BC, and BD data.

::<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\!</math>

Therefore, the demonstrated MTBF is:

::<math>MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\!</math>

Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:

::<math>\begin{align}
{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\
= & 0.09339
\end{align}\,\!</math>

::<math>\begin{align}
{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\
= & 0.17390
\end{align}\,\!</math>

The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:

::<math>\begin{align}
MTB{{F}_{{{D}_{L}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{U}}} \\
= & 5.75054 \\
MTB{{F}_{{{D}_{U}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{L}}} \\
= & 10.70799
\end{align}\,\!</math>

The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:

::<math>\begin{align}
MTB{{F}_{{{D}_{L}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{U}}} \\
= & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,\alpha /2)}{2N}} \\
= & 5.6325 \\
MTB{{F}_{{{D}_{U}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{L}}} \\
= & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,1-\alpha /2)}{2N}} \\
= & 10.8779
\end{align}\,\!</math>

The projected failure intensity is:

::<math>\begin{align}
\hat{\lambda}_P (T) &= \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} + \bar{d}\widehat{h}(T|BD) \\
&= 0.0885 \,\!
\end{align}</math>

Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:

::<math>\begin{align}
{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
= & 0.0681
\end{align}\,\!</math>

::<math>\begin{align}
{{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
= & 0.1152
\end{align}\,\!</math>

The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:

::<math>\begin{align}
MTB{{F}_{{{P}_{L}}}} = & \frac{1}{{{[{{\lambda }_{P}}(T)]}_{U}}} \\
= & 8.6818 \\
MTB{{F}_{{{P}_{U}}}} = & \frac{1}{{{[{{\lambda }_{P}}(T)]}_{L}}} \\
= & 14.6926
\end{align}\,\!</math>

The Crow confidence bounds for the projected failure intensity are:

::<math>\begin{align}
{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.0672 \\
{{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.1166
\end{align}\,\!</math>

The Crow confidence bounds for the projected MTBF at the 90% confidence level are:

::<math>\begin{align}
MTB{{F}_{{{P}_{L}}}} = & \frac{1}{{{[{{\widehat{\lambda }}_{P}}(T)]}_{U}}} \\
= & 8.5743 \\
MTB{{F}_{{{P}_{U}}}} = & \frac{1}{{{[{{\widehat{\lambda }}_{P}}(T)]}_{L}}} \\
= & 14.8769
\end{align}\,\!</math>

The growth potential failure intensity is:

::<math>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\!</math>

Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:

::<math>\begin{align}
{{r}_{L}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.0488 \\
{{r}_{U}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.0919
\end{align}\,\!</math>

The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:

::<math>\begin{align}
MTB{{F}_{G{{P}_{L}}}} = & \frac{1}{{{r}_{U}}} \\
= & 10.8790 \\
MTB{{F}_{G{{P}_{U}}}} = & \frac{1}{{{r}_{L}}} \\
= & 20.4855
\end{align}\,\!</math>

The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.

[[Image:rga9.10.png|center|400px|Fisher Matrix confidence bounds on MTBF.]]

The next figure shows these bounds based on the Crow method.

[[Image:rga9.11.png|center|400px|Crow confidence bounds on MTBF.]]