Crow Extended Confidence Bounds Example - Revision history

Lisa Hacker at 21:29, 18 September 2023

2023-09-18T21:29:47Z

← Older revision		Revision as of 21:29, 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 17:33, 30 May 2014

2014-05-30T17:33:12Z

← Older revision		Revision as of 17:33, 30 May 2014
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	The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</math>. Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:		The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</math>. 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 14:		Line 13:
	= & 0.08152		= & 0.08152
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 20:		Line 18:
	= & 0.13525		= & 0.13525
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
Line 31:		Line 27:
	= & 0.13299		= & 0.13299
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
Line 40:		Line 34:
	&= 0.06611		&= 0.06611
	\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 49:		Line 41:
	= & 0.04902		= & 0.04902
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 55:		Line 46:
	= & 0.08915		= & 0.08915
	\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 66:		Line 55:
	= & 0.09090		= & 0.09090
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</math>.		:<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</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 83:		Line 68:
	= & 0.0656		= & 0.0656
	\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 failure intensity.		The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity.

	[[Image:rga9.8.png\|center\|~~400px\|Fisher Matrix confidence bounds for the failure intensity.~~]]		[[Image:rga9.8.png\|center\|450px]]

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

	[[Image:rga9.9.png\|center\|~~400px\|Crow confidence bounds for the failure intensity.~~]]		[[Image:rga9.9.png\|center\|450px]]

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

2014-04-22T23:19:30Z

← Older revision		Revision as of 23:19, 22 April 2014
Line 1:		Line 1:
	<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 23:57, 29 January 2014

2014-01-29T23:57:37Z

← Older revision		Revision as of 23:57, 29 January 2014
Line 8:		Line 8:

	The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</math>. Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:		The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</math>. 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 13:		Line 14:
	= & 0.08152		= & 0.08152
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 18:		Line 20:
	= & 0.13525		= & 0.13525
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
Line 27:		Line 31:
	= & 0.13299		= & 0.13299
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
Line 37:		Line 43:

	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 42:		Line 49:
	= & 0.04902		= & 0.04902
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
Line 47:		Line 55:
	= & 0.08915		= & 0.08915
	\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 56:		Line 66:
	= & 0.09090		= & 0.09090
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</math>.		:<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</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 69:		Line 83:
	= & 0.0656		= & 0.0656
	\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 failure intensity.		The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity.

David J. Groebel at 18:11, 27 January 2014

2014-01-27T18:11:39Z

← Older revision		Revision as of 18:11, 27 January 2014
Line 4:		Line 4:

	Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the Test-Find-Test data <noinclude>shown in the [[Test-Find-Test_Data_Example\|Crow Extended Test-Find-Test Data example]]</noinclude><includeonly>given above</includeonly>.		Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the Test-Find-Test data <noinclude>shown in the [[Test-Find-Test_Data_Example\|Crow Extended Test-Find-Test Data example]]</noinclude><includeonly>given above</includeonly>.


	'''Solution'''		'''Solution'''
Line 10:		Line 9:
	The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</math>. Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:		The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</math>. 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}
	{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\		{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
	= & 0.08152		= & 0.08152
	\end{align}\,\!</math>		\end{align}\,\!</math>

	::<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\		{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
	= & 0.13525		= & 0.13525
	\end{align}\,\!</math>		\end{align}\,\!</math>


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

	::<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{D}}(T)]}_{L}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\		{{[{{\lambda }_{D}}(T)]}_{L}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\
	= & 0.07985 \\		= & 0.07985 \\
Line 29:		Line 27:
	= & 0.13299		= & 0.13299
	\end{align}\,\!</math>		\end{align}\,\!</math>


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

	::<math>\begin{align}		:<math>\begin{align}
	\hat{\lambda_{p}} &= \frac{N_{i}}{T}+\sum_{i=1}^{M}(1-d_{i})\frac{N}{T}+\overline{d}\left(\frac{M}{T}\overline{\beta} \right )\\		\hat{\lambda_{p}} &= \frac{N_{i}}{T}+\sum_{i=1}^{M}(1-d_{i})\frac{N}{T}+\overline{d}\left(\frac{M}{T}\overline{\beta} \right )\\
	&= 0.06611		&= 0.06611
Line 41:		Line 38:
	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}
	{{[{{{\hat{\lambda }}}_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\		{{[{{{\hat{\lambda }}}_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
	= & 0.04902		= & 0.04902
	\end{align}\,\!</math>		\end{align}\,\!</math>

	::<math>\begin{align}		:<math>\begin{align}
	{{[{{{\hat{\lambda }}}_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\		{{[{{{\hat{\lambda }}}_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
	= & 0.08915		= & 0.08915
	\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}
	{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\		{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
	= & 0.04807 \\		= & 0.04807 \\
Line 60:		Line 56:
	= & 0.09090		= & 0.09090
	\end{align}\,\!</math>		\end{align}\,\!</math>


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

	::<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</math>.		:<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</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}
	{{r}_{L}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\		{{r}_{L}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
	= & 0.03020 \\		= & 0.03020 \\
Line 75:		Line 69:
	= & 0.0656		= & 0.0656
	\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 failure intensity.		The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity.

	[[Image:rga9.8.png\|center\|400px\|Fisher Matrix confidence bounds for the failure intensity.]]		[[Image:rga9.8.png\|center\|400px\|Fisher Matrix confidence bounds for the failure intensity.]]


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

	[[Image:rga9.9.png\|center\|400px\|Crow confidence bounds for the failure intensity.]]		[[Image:rga9.9.png\|center\|400px\|Crow confidence bounds for the failure intensity.]]

Kate Racaza at 19:46, 20 January 2014

2014-01-20T19:46:09Z

← Older revision		Revision as of 19:46, 20 January 2014
Line 3:		Line 3:
	</noinclude>		</noinclude>

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

Kate Racaza at 19:32, 20 January 2014

2014-01-20T19:32:21Z

← Older revision		Revision as of 19:32, 20 January 2014
Line 3:		Line 3:
	</noinclude>		</noinclude>

	Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the ~~test~~-~~find~~-~~test~~ data <noinclude>shown in the [[Crow_Extended#Example:_Test-Find-Test_Data\|Crow Extended Test-Find-Test Data example]]</noinclude><includeonly>given above</includeonly>.		Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the Test-Find-Test data <noinclude>shown in the [[Crow_Extended#Example:_Test-Find-Test_Data\|Crow Extended Test-Find-Test Data example]]</noinclude><includeonly>given above</includeonly>.

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

2014-01-20T19:30:50Z

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 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the test-find-test data <noinclude>shown in the [[Crow_Extended#Example:_Test-Find-Test_Data|Crow Extended Test-Find-Test Data example]]</noinclude><includeonly>given above</includeonly>.

'''Solution'''

The estimated demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!</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 }}}_{D}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.08152
\end{align}\,\!</math>

::<math>\begin{align}
{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.13525
\end{align}\,\!</math>

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

::<math>\begin{align}
{{[{{\lambda }_{D}}(T)]}_{L}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\
= & 0.07985 \\
{{[{{\lambda }_{D}}(T)]}_{U}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N} \\
= & 0.13299
\end{align}\,\!</math>

The projected failure intensity is:

::<math>\begin{align}
\hat{\lambda_{p}} &= \frac{N_{i}}{T}+\sum_{i=1}^{M}(1-d_{i})\frac{N}{T}+\overline{d}\left(\frac{M}{T}\overline{\beta} \right )\\
&= 0.06611
\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}
{{[{{{\hat{\lambda }}}_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
= & 0.04902
\end{align}\,\!</math>

::<math>\begin{align}
{{[{{{\hat{\lambda }}}_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
= & 0.08915
\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}} \\
= & 0.04807 \\
{{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.09090
\end{align}\,\!</math>

The growth potential failure intensity is:

::<math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!</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}} \\
= & 0.03020 \\
{{r}_{U}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
= & 0.0656
\end{align}\,\!</math>

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

[[Image:rga9.8.png|center|400px|Fisher Matrix confidence bounds for the failure intensity.]]

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

[[Image:rga9.9.png|center|400px|Crow confidence bounds for the failure intensity.]]