Template:Characteristics of the Weibull Distribution - Revision history

Lisa Hacker: Redirected page to Weibull Distribution Characteristics

2012-08-07T02:50:42Z

Redirected page to Weibull Distribution Characteristics

← Older revision		Revision as of 02:50, 7 August 2012
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	~~== Characteristics of the Weibull Distribution ==~~		#REDIRECT [[Weibull Distribution Characteristics]]

	As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, <span class="texhtml">β</span>, and the scale parameter, <span class="texhtml">η</span>, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where <span class="texhtml">γ = 0,</span> or the 1-parameter form where <span class="texhtml">β = ''C'' = </span>constant, can easily be made.

	~~=== Characteristic Effects of the Shape Parameter, <span class="texhtml">β</span> ===~~

	The Weibull shape parameter, <span class="texhtml">β</span>, is also known as the ''slope''. This is because the value of <span class="texhtml">β</span> is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when <span class="texhtml">β = 1</span>, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

	~~::<math> f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} </math>~~

	~~where <math> \frac{1}{\eta }=\lambda = </math> failure rate. The parameter <span class="texhtml">β</span> is a pure number, i.e. it is dimensionless.~~


	~~{{The effect of beta on the Weibull pdf}}~~


	~~{{effect of beta on the cdf}}~~

	~~{{The effect of beta on Weibull failure rate}}~~

	~~=== Characteristic Effects of the Scale Parameter, <span class="texhtml">η</span> ===~~

	[[~~Image:WB.8 effects of n.png\|center\|250px\| The effects of <math>\eta</math> on the~~ Weibull ~~<math>pdf</math> for a common <math>\beta</math>. ]]~~


	A change in the scale parameter <span class="texhtml">η</span> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <span class="texhtml">η</span> while holding <span class="texhtml">β</span> constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of <span class="texhtml">η</span>, as indicated in the above figure.

	:*If <span class="texhtml">η</span> is increased while <span class="texhtml">β</span> and <span class="texhtml">γ</span> are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
	:*If <span class="texhtml">η</span> is decreased while <span class="texhtml">β</span> and <span class="texhtml">γ</span> are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or <span class="texhtml">γ</span>), and its height increases.
	~~:*<span class="texhtml">η</span> has the same units as , such as hours, miles, cycles, actuations, etc.~~

	~~<br>~~

	~~=== Characteristic Effects of the Location Parameter, <span class="texhtml">γ</span> ===~~

	The location parameter, <span class="texhtml">γ</span>, as the name implies, locates the distribution along the abscissa. Changing the value of <span class="texhtml">γ</span> has the effect of ''sliding'' the distribution and its associated function either to the right (if <span class="texhtml">γ > 0</span>) or to the left (if <span class="texhtml">γ < 0</span>).''


	~~[[Image:WB.8 location parameter.png\|center\|250px\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull <math>pdf</math>.~~ ]]

	~~:*When <span class="texhtml">γ = 0,</span> the distribution starts at or at the origin.~~
	~~:*If <span class="texhtml">γ > 0,</span> the distribution starts at the location <span class="texhtml">γ</span> to the right of the origin.~~
	~~:*If <span class="texhtml">γ < 0,</span> the distribution starts at the location <span class="texhtml">γ</span> to the left of the origin.~~
	~~:*<span class="texhtml">γ</span> provides an estimate of the earliest time-to-failure of such units.~~
	~~:*The life period 0 to <span class="texhtml">+ γ</span> is a failure free operating period of such units.~~
	:*The parameter <span class="texhtml">γ</span> may assume all values and provides an estimate of the earliest time a failure may be observed. A negative <span class="texhtml">γ</span> may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
	~~:*<span class="texhtml">γ</span> has the same units as T, such as hours, miles, cycles, actuations, etc.~~

Nicolette Young: /* Characteristic Effects of the Location Parameter, γ */

2012-04-25T18:02:09Z

Characteristic Effects of the Location Parameter, γ

← Older revision		Revision as of 18:02, 25 April 2012
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	[[Image:WB.8 location parameter.png\|center\|~~400px~~\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull <math>pdf</math>. ]]		[[Image:WB.8 location parameter.png\|center\|250px\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull <math>pdf</math>. ]]

	:*When <span class="texhtml">γ = 0,</span> the distribution starts at or at the origin.		:*When <span class="texhtml">γ = 0,</span> the distribution starts at or at the origin.

Nicolette Young: /* Characteristic Effects of the Scale Parameter, η */

2012-04-25T18:01:57Z

Characteristic Effects of the Scale Parameter, η

← Older revision		Revision as of 18:01, 25 April 2012
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	=== Characteristic Effects of the Scale Parameter, <span class="texhtml">η</span> ===		=== Characteristic Effects of the Scale Parameter, <span class="texhtml">η</span> ===

	[[Image:WB.8 effects of n.png\|center\|~~400px~~\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]		[[Image:WB.8 effects of n.png\|center\|250px\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]

Chris Kahn: /* Characteistic Effects of the Scale Parameter, η */

2012-04-24T22:59:49Z

Characteistic Effects of the Scale Parameter, η

← Older revision		Revision as of 22:59, 24 April 2012
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	{{The effect of beta on Weibull failure rate}}		{{The effect of beta on Weibull failure rate}}

	=== ~~Characteistic~~ Effects of the Scale Parameter, <span class="texhtml">η</span> ===		=== Characteristic Effects of the Scale Parameter, <span class="texhtml">η</span> ===

	[[Image:WB.8 effects of n.png\|center\|400px\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]		[[Image:WB.8 effects of n.png\|center\|400px\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]

Chris Kahn: /* Characteristics of the Weibull Distribution */

2012-04-24T22:57:17Z

Characteristics of the Weibull Distribution

← Older revision		Revision as of 22:57, 24 April 2012
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	== Characteristics of the Weibull Distribution ==		== Characteristics of the Weibull Distribution ==

	As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, <span class="texhtml">β</span>, and the scale parameter, <span class="texhtml">η</span>, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the ~~three~~-parameter form. The appropriate substitutions to obtain the other forms, such as the ~~two~~-parameter form where <span class="texhtml">γ = 0,</span> or the ~~one~~-parameter form where <span class="texhtml">β = ''C'' = </span>constant, can easily be made.		As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, <span class="texhtml">β</span>, and the scale parameter, <span class="texhtml">η</span>, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where <span class="texhtml">γ = 0,</span> or the 1-parameter form where <span class="texhtml">β = ''C'' = </span>constant, can easily be made.

	=== Characteristic Effects of the Shape Parameter, <span class="texhtml">β</span> ===		=== Characteristic Effects of the Shape Parameter, <span class="texhtml">β</span> ===

	The Weibull shape parameter, <span class="texhtml">β</span>, is also known as the ''slope''. This is because the value of <span class="texhtml">β</span> is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when <span class="texhtml">β = 1</span>, the of the ~~three~~-parameter Weibull reduces to that of the ~~two~~-parameter exponential distribution or:		The Weibull shape parameter, <span class="texhtml">β</span>, is also known as the ''slope''. This is because the value of <span class="texhtml">β</span> is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when <span class="texhtml">β = 1</span>, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

	::<math> f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} </math>		::<math> f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} </math>

Nicolette Young: /* Characteristic Effects of the Location Parameter, γ */

2012-03-15T16:39:51Z

Characteristic Effects of the Location Parameter, γ

← Older revision		Revision as of 16:39, 15 March 2012
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	[[Image:~~lda6~~.6.~~gif\|thumb~~\|center\|~~300px~~\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull <math>pdf</math>. ]]		[[Image:WB.8 location parameter.png\|center\|400px\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull <math>pdf</math>. ]]

	:*When <span class="texhtml">γ = 0,</span> the distribution starts at or at the origin.		:*When <span class="texhtml">γ = 0,</span> the distribution starts at or at the origin.

Nicolette Young: /* Characteistic Effects of the Scale Parameter, η */

2012-03-15T16:39:21Z

Characteistic Effects of the Scale Parameter, η

← Older revision		Revision as of 16:39, 15 March 2012
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	=== Characteistic Effects of the Scale Parameter, <span class="texhtml">η</span> ===		=== Characteistic Effects of the Scale Parameter, <span class="texhtml">η</span> ===

	[[Image:~~lda6~~.5.~~gif\|thumb~~\|center\|~~300px~~\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]		[[Image:WB.8 effects of n.png\|center\|400px\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]

Harry Guo: /* Characteristics of the Weibull Distribution */

2012-02-09T22:01:09Z

Characteristics of the Weibull Distribution

← Older revision		Revision as of 22:01, 9 February 2012
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	where <math> \frac{1}{\eta }=\lambda = </math> failure rate. The parameter <span class="texhtml">β</span> is a pure number, i.e. it is dimensionless.		where <math> \frac{1}{\eta }=\lambda = </math> failure rate. The parameter <span class="texhtml">β</span> is a pure number, i.e. it is dimensionless.


	{{The effect of beta on the Weibull pdf}}		{{The effect of beta on the Weibull pdf}}


	{{effect of beta on the cdf}}		{{effect of beta on the cdf}}

Harry Guo: /* Characteristics of the Weibull Distribution */

2012-02-09T22:00:39Z

Characteristics of the Weibull Distribution

← Older revision		Revision as of 22:00, 9 February 2012
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	{{The effect of beta on Weibull failure rate}}		{{The effect of beta on Weibull failure rate}}

			=== Characteistic Effects of the Scale Parameter, <span class="texhtml">η</span> ===
	=== ~~Characteristic~~ Effects of the Scale Parameter, <span class="texhtml">η</span> ===


	[[Image:lda6.5.gif\|thumb\|center\|300px\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]		[[Image:lda6.5.gif\|thumb\|center\|300px\| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]


	A change in the scale parameter <span class="texhtml">η</span> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <span class="texhtml">η</span> while holding <span class="texhtml">β</span> constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of <span class="texhtml">η</span>, as indicated in ~~Figure 6-5~~.		A change in the scale parameter <span class="texhtml">η</span> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <span class="texhtml">η</span> while holding <span class="texhtml">β</span> constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of <span class="texhtml">η</span>, as indicated in the above figure.

	:*If <span class="texhtml">η</span> is increased while <span class="texhtml">β</span> and <span class="texhtml">γ</span> are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.		:*If <span class="texhtml">η</span> is increased while <span class="texhtml">β</span> and <span class="texhtml">γ</span> are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.

Harry Guo: /* Characteristics of the Weibull Distribution */

2012-02-09T21:59:39Z

Characteristics of the Weibull Distribution

← Older revision		Revision as of 21:59, 9 February 2012
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	{{The effect of beta on Weibull failure rate}}		{{The effect of beta on Weibull failure rate}}