Template:ReliabilityFunction - Revision history

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-07-20T12:09:12Z

Replaced content with 'Category: For Deletion'

← Older revision		Revision as of 12:09, 20 July 2012
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	~~==='''The Reliability Function'''===~~		[[Category: For Deletion]]

	~~The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:~~

	~~::<math>F(t)=\int_{0,\gamma}^{t}f(s)ds</math>~~

	~~In particular, this represents the probability of a unit failing by time <math>t</math>.~~
	From this, we obtain the most commonly used function in reliability engineering, the reliability function, which represents the probability of success of a unit in undertaking a mission of a prescribed duration.
	To mathematically show this, we first define the unreliability function, <math>Q(t)</math>, which is the probability of failure, or the probability that our time-to-failure is in the region of <math>0</math> (or <math>\gamma </math>) and <math>t</math>. So from the previous equation:

	~~::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>~~

	[[~~Image~~:~~reliabilityfunction.png\|center\|200px\|~~]]

	In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:

	~~::<math>\begin{align}~~
	~~Q(t)+R(t)& = 1 \\~~
	~~R(t) & = 1-Q(t) \\~~
	~~R(t) & = 1-\int_{0,\gamma}^{t}f(s)ds \\~~
	~~R(t) & = \int_{t}^{\infty }f(s)ds~~
	~~\end{align}</math>~~

	~~Conversely:~~

	~~::<math>f(t)=-\frac{d(R(t))}{dt}</math>~~

Nicolette Young: /* The Reliability Function */

2012-04-25T16:43:39Z

The Reliability Function

← Older revision		Revision as of 16:43, 25 April 2012
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	::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>		::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>

	[[Image:reliabilityfunction.png\|center\|~~400px~~\|]]		[[Image:reliabilityfunction.png\|center\|200px\|]]

	In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:		In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:

Nicolette Young: /* The Reliability Function */

2012-03-12T16:26:49Z

The Reliability Function

← Older revision		Revision as of 16:26, 12 March 2012
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	::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>		::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>

	[[Image:reliabilityfunction.png~~\|thumb~~\|center\|400px\|]]		[[Image:reliabilityfunction.png\|center\|400px\|]]

	In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:		In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:

Nicolette Young: /* The Reliability Function */

2012-03-12T16:26:13Z

The Reliability Function

← Older revision		Revision as of 16:26, 12 March 2012
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	::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>		::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>

	[[Image:~~prob-of-fail~~.png\|thumb\|center\|400px\|]]		[[Image:reliabilityfunction.png\|thumb\|center\|400px\|]]

	In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:		In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:

Harry Guo: /* The Reliability Function */

2012-02-08T21:09:35Z

The Reliability Function

← Older revision		Revision as of 21:09, 8 February 2012
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	=='''The Reliability Function'''==		==='''The Reliability Function'''===

	The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:		The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:

Harry Guo at 21:04, 8 February 2012

2012-02-08T21:04:00Z

← Older revision		Revision as of 21:04, 8 February 2012
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	==='''The Reliability Function'''===		=='''The Reliability Function'''==

	The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:		The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:

Harry Guo: /* The Reliability Function */

2012-02-06T16:19:44Z

The Reliability Function

← Older revision		Revision as of 16:19, 6 February 2012
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	\end{align}</math>		\end{align}</math>

	:Conversely:		Conversely:

	::<math>f(t)=-\frac{d(R(t))}{dt}</math>		::<math>f(t)=-\frac{d(R(t))}{dt}</math>

Nicolette Young at 22:17, 3 January 2012

2012-01-03T22:17:23Z

← Older revision		Revision as of 22:17, 3 January 2012
Line 1:		Line 1:
			==='''The Reliability Function'''===

	The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:		The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:

Nicolette Young: Created page with 'The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time $t$ is giv…'

2012-01-03T22:10:49Z

Created page with 'The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is giv…'

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The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time <math>t</math> is given by:

::<math>F(t)=\int_{0,\gamma}^{t}f(s)ds</math>

In particular, this represents the probability of a unit failing by time <math>t</math>.
From this, we obtain the most commonly used function in reliability engineering, the reliability function, which represents the probability of success of a unit in undertaking a mission of a prescribed duration.
To mathematically show this, we first define the unreliability function, <math>Q(t)</math>, which is the probability of failure, or the probability that our time-to-failure is in the region of <math>0</math> (or <math>\gamma </math>) and <math>t</math>. So from the previous equation:

::<math>F(t)=Q(t)=\int_{0,\gamma}^{t}f(s)ds</math>

[[Image:prob-of-fail.png|thumb|center|400px|]]

In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:

::<math>\begin{align}
Q(t)+R(t)& = 1 \\
R(t) & = 1-Q(t) \\
R(t) & = 1-\int_{0,\gamma}^{t}f(s)ds \\
R(t) & = \int_{t}^{\infty }f(s)ds
\end{align}</math>

:Conversely:

::<math>f(t)=-\frac{d(R(t))}{dt}</math>

Template:ReliabilityFunction - Revision history

Lisa Hacker: Replaced content with 'Category: For Deletion'

Nicolette Young: /* The Reliability Function */

Nicolette Young: /* The Reliability Function */

Nicolette Young: /* The Reliability Function */

Harry Guo: /* The Reliability Function */

Harry Guo at 21:04, 8 February 2012

Harry Guo: /* The Reliability Function */

Nicolette Young at 22:17, 3 January 2012

Nicolette Young: Created page with 'The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time t is giv…'

Nicolette Young: Created page with 'The reliability function can be derived using the previous definition of the cumulative distribution function. The probability of an event happening by time $t$ is giv…'