Template:Characteristics of the Exponential Distribution - Revision history

Lisa Hacker: Redirected page to Exponential Distribution Characteristics

2012-08-08T07:42:17Z

Redirected page to Exponential Distribution Characteristics

← Older revision		Revision as of 07:42, 8 August 2012
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	~~==Characteristics of the Exponential Distribution==~~		#REDIRECT [[Exponential Distribution Characteristics]]
	As mentioned before, the primary trait of the exponential distribution is that it is used for modeling the behavior of items with a constant failure rate. It has a fairly simple mathematical form, which makes it fairly easy to manipulate. Unfortunately, this fact also leads to the use of this model in situations where it is not appropriate. For example, it would not be appropriate to use the exponential distribution to model the reliability of an automobile. The constant failure rate of the exponential distribution would require the assumption that the automobile would be just as likely to experience a breakdown during the first mile as it would during the one-hundred-thousandth mile. Clearly, this is not a valid assumption. However, some inexperienced practitioners of reliability engineering and life data analysis will overlook this fact, lured by the siren-call of the exponential distribution's relatively simple mathematical models.

	~~===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential <math>pdf</math>===~~

	[[~~Image:effect of lambda on exponential pdf.png\|center\|250px\|]]~~

	~~:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.~~
	~~:*The exponential <math>pdf</math> is always convex and is stretched to the right as <math>\lambda </math> decreases in value.~~
	~~:*The value of the <math>pdf</math> function is always equal to the value of <math>\lambda </math> at <math>t=0</math> (or <math>t=\gamma </math>).~~
	:*The location parameter, <math>\gamma </math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma </math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma </math> hours of operation, and cannot occur before this time.
	~~:*The scale parameter is <math>\tfrac{1}{\lambda }=\bar{T}-\gamma =m-\gamma </math>.~~
	~~:*As <math>t\to \infty </math>, <math>f(t)\to 0</math>.~~

	~~===The Effect of <math>\lambda </math> and <math>\gamma </math> on the~~ Exponential ~~Reliability Function===~~

	~~[[Image:effect of upsilon.png\|center\|250px\|~~]]

	~~:*The one-parameter exponential reliability function starts at the value of 100% at <math>t=0</math>, decreases thereafter monotonically and is convex.~~
	~~:*The two-parameter exponential reliability function remains at the value of 100% for <math>t=0</math> up to <math>t=\gamma </math>, and decreases thereafter monotonically and is convex.~~
	~~:*As <math>t\to \infty </math> , <math>R(t\to \infty )\to 0</math>.~~
	:*The reliability for a mission duration of <math>t=m=\tfrac{1}{\lambda }</math>, or of one MTTF duration, is always equal to <math>0.3679</math> or 36.79%. This means that the reliability for a mission which is as long as one MTTF is relatively low and is not recommended because only 36.8% of the missions will be completed successfully. In other words, of the equipment undertaking such a mission, only 36.8% will survive their mission.

Nicolette Young: /* The Effect of \lambda and \gamma on the Exponential Reliability Function */

2012-04-25T17:52:59Z

The Effect of \lambda and \gamma on the Exponential Reliability Function

← Older revision		Revision as of 17:52, 25 April 2012
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	===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===		===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===

	[[Image:effect of upsilon.png\|center\|~~400px~~\|]]		[[Image:effect of upsilon.png\|center\|250px\|]]

	:*The one-parameter exponential reliability function starts at the value of 100% at <math>t=0</math>, decreases thereafter monotonically and is convex.		:*The one-parameter exponential reliability function starts at the value of 100% at <math>t=0</math>, decreases thereafter monotonically and is convex.

Nicolette Young: /* The Effect of \lambda and \gamma on the Exponential pdf */

2012-04-25T17:52:50Z

The Effect of \lambda and \gamma on the Exponential pdf

← Older revision		Revision as of 17:52, 25 April 2012
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	===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential <math>pdf</math>===		===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential <math>pdf</math>===

	[[Image:effect of lambda on exponential pdf.png\|center\|~~400px~~\|]]		[[Image:effect of lambda on exponential pdf.png\|center\|250px\|]]

	:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.		:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.

Nicolette Young: /* The Effect of \lambda and \gamma on the Exponential Reliability Function */

2012-03-13T16:37:07Z

The Effect of \lambda and \gamma on the Exponential Reliability Function

← Older revision		Revision as of 16:37, 13 March 2012
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	===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===		===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===

	[[Image:~~effectofgammaonerf~~.~~gif\|thumb~~\|center\|~~300px~~\|]]		[[Image:effect of upsilon.png\|center\|400px\|]]

	:*The one-parameter exponential reliability function starts at the value of 100% at <math>t=0</math>, decreases thereafter monotonically and is convex.		:*The one-parameter exponential reliability function starts at the value of 100% at <math>t=0</math>, decreases thereafter monotonically and is convex.

Nicolette Young: /* The Effect of \lambda and \gamma on the Exponential pdf */

2012-03-13T16:35:24Z

The Effect of \lambda and \gamma on the Exponential pdf

← Older revision		Revision as of 16:35, 13 March 2012
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	===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential <math>pdf</math>===		===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential <math>pdf</math>===

	[[Image:~~effectoflambdaonpdf~~.~~gif\|thumb~~\|center\|~~300px~~\|]]		[[Image:effect of lambda on exponential pdf.png\|center\|400px\|]]

	:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.		:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.

Harry Guo at 23:16, 7 February 2012

2012-02-07T23:16:44Z

← Older revision		Revision as of 23:16, 7 February 2012
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	[[Image:effectofgammaonerf.gif\|thumb\|center\|300px\|]]		[[Image:effectofgammaonerf.gif\|thumb\|center\|300px\|]]

	:*The one-parameter exponential reliability function starts at the value of 100% at <math>T=0</math>, decreases thereafter monotonically and is convex.		:*The one-parameter exponential reliability function starts at the value of 100% at <math>t=0</math>, decreases thereafter monotonically and is convex.
	:*The two-parameter exponential reliability function remains at the value of 100% for <math>T=0</math> up to <math>T=\gamma </math>, and decreases thereafter monotonically and is convex.		:*The two-parameter exponential reliability function remains at the value of 100% for <math>t=0</math> up to <math>t=\gamma </math>, and decreases thereafter monotonically and is convex.
	:*As <math>T\to \infty </math> , <math>R(T\to \infty )\to 0</math>.		:*As <math>t\to \infty </math> , <math>R(t\to \infty )\to 0</math>.
	:*The reliability for a mission duration of <math>T=m=\tfrac{1}{\lambda }</math>, or of one MTTF duration, is always equal to <math>0.3679</math> or 36.79%. This means that the reliability for a mission which is as long as one MTTF is relatively low and is not recommended because only 36.8% of the missions will be completed successfully. In other words, of the equipment undertaking such a mission, only 36.8% will survive their mission.		:*The reliability for a mission duration of <math>t=m=\tfrac{1}{\lambda }</math>, or of one MTTF duration, is always equal to <math>0.3679</math> or 36.79%. This means that the reliability for a mission which is as long as one MTTF is relatively low and is not recommended because only 36.8% of the missions will be completed successfully. In other words, of the equipment undertaking such a mission, only 36.8% will survive their mission.

Harry Guo: /* The Effect of \lambda and \gamma on the Exponential pdf */

2012-02-07T23:15:35Z

The Effect of \lambda and \gamma on the Exponential pdf

← Older revision		Revision as of 23:15, 7 February 2012
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	:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.		:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.
	:*The exponential <math>pdf</math> is always convex and is stretched to the right as <math>\lambda </math> decreases in value.		:*The exponential <math>pdf</math> is always convex and is stretched to the right as <math>\lambda </math> decreases in value.
	:*The value of the <math>pdf</math> function is always equal to the value of <math>\lambda </math> at <math>T=0</math> (or <math>T=\gamma </math>).		:*The value of the <math>pdf</math> function is always equal to the value of <math>\lambda </math> at <math>t=0</math> (or <math>t=\gamma </math>).
	:*The location parameter, <math>\gamma </math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma </math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma </math> hours of operation, and cannot occur before this time.		:*The location parameter, <math>\gamma </math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma </math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma </math> hours of operation, and cannot occur before this time.
	:*The scale parameter is <math>\tfrac{1}{\lambda }=\bar{T}-\gamma =m-\gamma </math>.		:*The scale parameter is <math>\tfrac{1}{\lambda }=\bar{T}-\gamma =m-\gamma </math>.
	:*As <math>T\to \infty </math>, <math>f(T)\to 0</math>.		:*As <math>t\to \infty </math>, <math>f(t)\to 0</math>.

	===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===		===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===

Harry Guo: /* The Effect of \lambda and \gamma on the Exponential Reliability Function */

2012-02-07T23:01:09Z

The Effect of \lambda and \gamma on the Exponential Reliability Function

← Older revision		Revision as of 23:01, 7 February 2012
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	:*As <math>T\to \infty </math>, <math>f(T)\to 0</math>.		:*As <math>T\to \infty </math>, <math>f(T)\to 0</math>.

	====The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function====		===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function===

	[[Image:effectofgammaonerf.gif\|thumb\|center\|300px\|]]		[[Image:effectofgammaonerf.gif\|thumb\|center\|300px\|]]

Nicolette Young: Created page with '==Characteristics of the Exponential Distribution== As mentioned before, the primary trait of the exponential distribution is that it is used for modeling the behavior of items w…'

2012-01-04T15:50:21Z

Created page with '==Characteristics of the Exponential Distribution== As mentioned before, the primary trait of the exponential distribution is that it is used for modeling the behavior of items w…'

New page

==Characteristics of the Exponential Distribution==
As mentioned before, the primary trait of the exponential distribution is that it is used for modeling the behavior of items with a constant failure rate. It has a fairly simple mathematical form, which makes it fairly easy to manipulate. Unfortunately, this fact also leads to the use of this model in situations where it is not appropriate. For example, it would not be appropriate to use the exponential distribution to model the reliability of an automobile. The constant failure rate of the exponential distribution would require the assumption that the automobile would be just as likely to experience a breakdown during the first mile as it would during the one-hundred-thousandth mile. Clearly, this is not a valid assumption. However, some inexperienced practitioners of reliability engineering and life data analysis will overlook this fact, lured by the siren-call of the exponential distribution's relatively simple mathematical models.

===The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential <math>pdf</math>===

[[Image:effectoflambdaonpdf.gif|thumb|center|300px|]]

:*The exponential <math>pdf</math> has no shape parameter, as it has only one shape.
:*The exponential <math>pdf</math> is always convex and is stretched to the right as <math>\lambda </math> decreases in value.
:*The value of the <math>pdf</math> function is always equal to the value of <math>\lambda </math> at <math>T=0</math> (or <math>T=\gamma </math>).
:*The location parameter, <math>\gamma </math>, if positive, shifts the beginning of the distribution by a distance of <math>\gamma </math> to the right of the origin, signifying that the chance failures start to occur only after <math>\gamma </math> hours of operation, and cannot occur before this time.
:*The scale parameter is <math>\tfrac{1}{\lambda }=\bar{T}-\gamma =m-\gamma </math>.
:*As <math>T\to \infty </math>, <math>f(T)\to 0</math>.

====The Effect of <math>\lambda </math> and <math>\gamma </math> on the Exponential Reliability Function====

[[Image:effectofgammaonerf.gif|thumb|center|300px|]]

:*The one-parameter exponential reliability function starts at the value of 100% at <math>T=0</math>, decreases thereafter monotonically and is convex.
:*The two-parameter exponential reliability function remains at the value of 100% for <math>T=0</math> up to <math>T=\gamma </math>, and decreases thereafter monotonically and is convex.
:*As <math>T\to \infty </math> , <math>R(T\to \infty )\to 0</math>.
:*The reliability for a mission duration of <math>T=m=\tfrac{1}{\lambda }</math>, or of one MTTF duration, is always equal to <math>0.3679</math> or 36.79%. This means that the reliability for a mission which is as long as one MTTF is relatively low and is not recommended because only 36.8% of the missions will be completed successfully. In other words, of the equipment undertaking such a mission, only 36.8% will survive their mission.