Template:Normal statistical properties - Revision history

Richard House: Redirected page to The Normal Distribution

2012-08-13T02:48:37Z

Redirected page to The Normal Distribution

← Older revision		Revision as of 02:48, 13 August 2012
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	~~==Normal Statistical Properties==~~		#REDIRECT [[The_Normal_Distribution]]

	~~{{normal mean median and mode}}~~

	~~{{normal standard deviation}}~~

	~~{{normal reliability function}}~~

	~~{{normal conditional reliability function}}~~

	~~{{normal reliable life}}~~

	~~{{normal failure rate function}}~~

Nicolette Young: /* The Normal Failure Rate Function */

2012-01-04T17:29:51Z

The Normal Failure Rate Function

← Older revision		Revision as of 17:29, 4 January 2012
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	{{normal reliable life}}		{{normal reliable life}}

	~~===The Normal Failure Rate Function===~~		{{normal failure rate function}}

	~~The instantaneous~~ normal failure rate ~~is given by:~~

	::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{T-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}~~</math>~~

Nicolette Young: /* The Normal Reliable Life */

2012-01-04T17:29:24Z

The Normal Reliable Life

← Older revision		Revision as of 17:29, 4 January 2012
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	{{normal conditional reliability function}}		{{normal conditional reliability function}}

	~~===The Normal Reliable Life===~~		{{normal reliable life}}

	~~Since there is no closed-form solution for the normal reliability function, there will also be no closed-form solution for the normal reliable life. To determine the~~ normal reliable life~~, one must solve:~~

	~~::<math>R(T)=\int_{T}^{\infty }\frac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}~~}}~~dt</math>~~
	~~::for <math>T</math> .~~

	===The Normal Failure Rate Function===		===The Normal Failure Rate Function===

Nicolette Young: /* The Normal Conditional Reliability Function */

2012-01-04T17:28:23Z

The Normal Conditional Reliability Function

← Older revision		Revision as of 17:28, 4 January 2012
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	{{normal reliability function}}		{{normal reliability function}}

	~~===The Normal Conditional Reliability Function===~~		{{normal conditional reliability function}}

	~~The~~ normal conditional reliability function ~~is given by:~~

	::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{T+t}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}~~dt}</math>~~

	~~Once again, the use of standard normal tables for the calculation of the normal conditional reliability is necessary, as there is no closed form solution.~~


	===The Normal Reliable Life===		===The Normal Reliable Life===

Nicolette Young: /* The Normal Reliability Function */

2012-01-04T17:27:50Z

The Normal Reliability Function

← Older revision		Revision as of 17:27, 4 January 2012
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	{{normal standard deviation}}		{{normal standard deviation}}

	~~===The Normal Reliability Function===~~		{{normal reliability function}}
	~~The reliability for a mission of time <math>T</math> for the normal distribution is determined by:~~

	~~::<math>R(T)=\int_{T}^{\infty }f(t)dt=\int_{T}^{\infty }\frac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}~~{2}{~~{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt</math>~~

	~~There is no closed-form solution for the~~ normal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability, we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.

	===The Normal Conditional Reliability Function===		===The Normal Conditional Reliability Function===

Nicolette Young: /* The Normal Standard Deviation */

2012-01-04T17:27:24Z

The Normal Standard Deviation

← Older revision		Revision as of 17:27, 4 January 2012
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	{{normal mean median and mode}}		{{normal mean median and mode}}

	~~===The Normal Standard Deviation===~~		{{normal standard deviation}}

	~~As with the mean, the~~ standard deviation ~~for the normal distribution is actually one of the parameters, usually denoted as <math>{{\sigma }_{T~~}}~~.</math>~~

	===The Normal Reliability Function===		===The Normal Reliability Function===

Nicolette Young: /* The Normal Mean, Median and Mode */

2012-01-04T17:26:12Z

The Normal Mean, Median and Mode

← Older revision		Revision as of 17:26, 4 January 2012
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	==Normal Statistical Properties==		==Normal Statistical Properties==

	~~===The Normal Mean, Median and Mode===~~		{{normal mean median and mode}}

	~~The~~ normal mean ~~or MTTF is actually one of the parameters of the distribution, usually denoted as <math>\mu .</math> Since the normal distribution is symmetrical, the~~ median and ~~the~~ mode ~~are always equal to the mean,~~

	~~::<math>\mu =\tilde{T~~}~~=\breve{T~~}~~</math>~~

	===The Normal Standard Deviation===		===The Normal Standard Deviation===

Nicolette Young: Created page with '==Normal Statistical Properties== ===The Normal Mean, Median and Mode=== The normal mean or MTTF is actually one of the parameters of the distribution, usually denoted as
2012-01-04T17:25:34Z

Created page with '==Normal Statistical Properties== ===The Normal Mean, Median and Mode=== The normal mean or MTTF is actually one of the parameters of the distribution, usually denoted as <mat…'

New page
==Normal Statistical Properties==

===The Normal Mean, Median and Mode===

The normal mean or MTTF is actually one of the parameters of the distribution, usually denoted as <math>\mu .</math> Since the normal distribution is symmetrical, the median and the mode are always equal to the mean,

::<math>\mu =\tilde{T}=\breve{T}</math>

===The Normal Standard Deviation===

As with the mean, the standard deviation for the normal distribution is actually one of the parameters, usually denoted as <math>{{\sigma }_{T}}.</math>

===The Normal Reliability Function===
The reliability for a mission of time <math>T</math> for the normal distribution is determined by:

::<math>R(T)=\int_{T}^{\infty }f(t)dt=\int_{T}^{\infty }\frac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt</math>

There is no closed-form solution for the normal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability, we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.

===The Normal Conditional Reliability Function===

The normal conditional reliability function is given by:

::<math>R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{T+t}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}</math>

Once again, the use of standard normal tables for the calculation of the normal conditional reliability is necessary, as there is no closed form solution.

===The Normal Reliable Life===

Since there is no closed-form solution for the normal reliability function, there will also be no closed-form solution for the normal reliable life. To determine the normal reliable life, one must solve:

::<math>R(T)=\int_{T}^{\infty }\frac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt</math>
::for <math>T</math> .

===The Normal Failure Rate Function===

The instantaneous normal failure rate is given by:

::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{T-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}</math>