Weibull Distribution Characteristics - Revision history

Lisa Hacker at 21:44, 18 September 2023

2023-09-18T21:44:13Z

← Older revision		Revision as of 21:44, 18 September 2023
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	<noinclude>{{Navigation box}}[[Category:Shared Articles]]		<noinclude>{{Navigation box}}[[Category:Shared Articles]]
	''This article also appears in the [~~[The_Weibull_Distribution\|~~Life ~~Data Analysis Reference]~~] and [~~[Distributions_Used_in_Accelerated_Testing\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] ~~books~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference] and [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference].'' </noinclude>

	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, <math>\beta\,\!</math>, and the scale parameter, <math>\eta\,\!</math>, 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 <math>\gamma = 0,\,\!</math> or the 1-parameter form where <math>\beta = C = \,\!</math> constant, can easily be made.		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, <math>\beta\,\!</math>, and the scale parameter, <math>\eta\,\!</math>, 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 <math>\gamma = 0,\,\!</math> or the 1-parameter form where <math>\beta = C = \,\!</math> constant, can easily be made.

Nicolette Young: /* Effects of the Shape Parameter, beta */

2015-12-17T15:15:26Z

Effects of the Shape Parameter, beta

← Older revision		Revision as of 15:15, 17 December 2015
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	'''The Effect of beta on the ''cdf'' and Reliability Function'''		'''The Effect of beta on the ''cdf'' and Reliability Function'''

	[[Image:WB.8 effect of weibull.png\|center\|~~450px~~\| Effect on <math>\beta\,\!</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta\,\!</math> ]]		[[Image:WB.8 effect of weibull.png\|center\|500px\| Effect on <math>\beta\,\!</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta\,\!</math> ]]

	The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.		The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.

	[[Image:WB.8 weibull reliability.png\|center\|~~450px~~\| The effect of values of <math>\beta\,\!</math> on the Weibull reliability plot. ]]		[[Image:WB.8 weibull reliability.png\|center\|500px\| The effect of values of <math>\beta\,\!</math> on the Weibull reliability plot. ]]

	:*<math>R(t)\,\!</math> decreases sharply and monotonically for <math>0 < \beta < 1\,\!</math> and is convex.		:*<math>R(t)\,\!</math> decreases sharply and monotonically for <math>0 < \beta < 1\,\!</math> and is convex.
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	The value of <math>\beta\,\!</math> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <math>\beta\,\!</math> is less than, equal to, or greater than one.		The value of <math>\beta\,\!</math> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <math>\beta\,\!</math> is less than, equal to, or greater than one.

	[[Image:WB.8 weibull failure.png\|center\|~~450px~~\| The effect of <math>\beta\,\!</math> on the Weibull failure rate function. ]]		[[Image:WB.8 weibull failure.png\|center\|500px\| The effect of <math>\beta\,\!</math> on the Weibull failure rate function. ]]

	As indicated by above figure, populations with <math>\beta < 1\,\!</math> exhibit a failure rate that decreases with time, populations with <math>\beta = 1\,\!</math> have a constant failure rate (consistent with the exponential distribution) and populations with <math>\beta > 1\,\!</math> have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <math>\beta\,\!</math>. The Weibull failure rate for <math>0 < \beta < 1\,\!</math> is unbounded at <math>T = 0\,\!</math> (or <math>\gamma\,\!)\,\!</math>. The failure rate, <math>\lambda(t),\,\!</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>t\rightarrow \infty\,\!</math> or <math>\lambda (\infty) = 0\,\!</math>. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <math>\beta = 1\,\!</math>, <math>\lambda(t)\,\!</math> yields a constant value of <math> { \frac{1}{\eta }} \,\!</math> or:		As indicated by above figure, populations with <math>\beta < 1\,\!</math> exhibit a failure rate that decreases with time, populations with <math>\beta = 1\,\!</math> have a constant failure rate (consistent with the exponential distribution) and populations with <math>\beta > 1\,\!</math> have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <math>\beta\,\!</math>. The Weibull failure rate for <math>0 < \beta < 1\,\!</math> is unbounded at <math>T = 0\,\!</math> (or <math>\gamma\,\!)\,\!</math>. The failure rate, <math>\lambda(t),\,\!</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>t\rightarrow \infty\,\!</math> or <math>\lambda (\infty) = 0\,\!</math>. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <math>\beta = 1\,\!</math>, <math>\lambda(t)\,\!</math> yields a constant value of <math> { \frac{1}{\eta }} \,\!</math> or:

Lisa Hacker: /* Effects of the Location Parameter, gamma */

2014-02-07T00:20:28Z

Effects of the Location Parameter, gamma

← Older revision		Revision as of 00:20, 7 February 2014
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	The location parameter, <math>\gamma\,\!</math>, as the name implies, locates the distribution along the abscissa. Changing the value of <math>\gamma\,\!</math> has the effect of ''sliding'' the distribution and its associated function either to the right (if <math>\gamma > 0\,\!</math>) or to the left (if <math>\gamma < 0\,\!</math>).		The location parameter, <math>\gamma\,\!</math>, as the name implies, locates the distribution along the abscissa. Changing the value of <math>\gamma\,\!</math> has the effect of ''sliding'' the distribution and its associated function either to the right (if <math>\gamma > 0\,\!</math>) or to the left (if <math>\gamma < 0\,\!</math>).

	[[Image:WB.8 location parameter.png\|center\|~~350px~~\| The effect of a positive location parameter, <math>\gamma\,\!</math>, on the position of the Weibull ''pdf''. ]]		[[Image:WB.8 location parameter.png\|center\|450px\| The effect of a positive location parameter, <math>\gamma\,\!</math>, on the position of the Weibull ''pdf''. ]]

	:*When <math>\gamma = 0,\,\!</math> the distribution starts at <math>t=0\,\!</math> or at the origin.		:*When <math>\gamma = 0,\,\!</math> the distribution starts at <math>t=0\,\!</math> or at the origin.

Lisa Hacker: /* Effects of the Scale Parameter, eta */

2014-02-07T00:20:17Z

Effects of the Scale Parameter, eta

← Older revision		Revision as of 00:20, 7 February 2014
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	=== Effects of the Scale Parameter, eta ===		=== Effects of the Scale Parameter, eta ===
	[[Image:WB.8 effects of n.png\|center\|~~350px~~\| The effects of <math>\eta\,\!</math> on the Weibull ''pdf'' for a common <math>\beta\,\!</math>. ]]		[[Image:WB.8 effects of n.png\|center\|450px\| The effects of <math>\eta\,\!</math> on the Weibull ''pdf'' for a common <math>\beta\,\!</math>. ]]

	A change in the scale parameter <math>\eta\,\!</math> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <math>\eta\,\!</math> while holding <math>\beta\,\!</math> constant has the effect of stretching out the ''pdf''. Since the area under a ''pdf'' curve is a constant value of one, the "peak" of the ''pdf'' curve will also decrease with the increase of <math>\eta\,\!</math>, as indicated in the above figure.		A change in the scale parameter <math>\eta\,\!</math> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <math>\eta\,\!</math> while holding <math>\beta\,\!</math> constant has the effect of stretching out the ''pdf''. Since the area under a ''pdf'' curve is a constant value of one, the "peak" of the ''pdf'' curve will also decrease with the increase of <math>\eta\,\!</math>, as indicated in the above figure.

Lisa Hacker: /* Effects of the Shape Parameter, beta */

2014-02-07T00:18:33Z

Effects of the Shape Parameter, beta

← Older revision		Revision as of 00:18, 7 February 2014
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	The following figure shows the effect of different values of the shape parameter, <math>\beta\,\!</math>, on the shape of the ''pdf''. As you can see, the shape can take on a variety of forms based on the value of <math>\beta\,\!</math>.		The following figure shows the effect of different values of the shape parameter, <math>\beta\,\!</math>, on the shape of the ''pdf''. As you can see, the shape can take on a variety of forms based on the value of <math>\beta\,\!</math>.

	[[Image:WB.8 weibull pdf.png\|center\|~~400px~~\| The effect of the Weibull shape parameter on the ''pdf''.]]		[[Image:WB.8 weibull pdf.png\|center\|450px\| The effect of the Weibull shape parameter on the ''pdf''.]]

	For <math> 0<\beta \leq 1 \,\!</math>:		For <math> 0<\beta \leq 1 \,\!</math>:
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	'''The Effect of beta on the ''cdf'' and Reliability Function'''		'''The Effect of beta on the ''cdf'' and Reliability Function'''

	[[Image:WB.8 effect of weibull.png\|center\|~~400px~~\| Effect on <math>\beta\,\!</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta\,\!</math> ]]		[[Image:WB.8 effect of weibull.png\|center\|450px\| Effect on <math>\beta\,\!</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta\,\!</math> ]]

	The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.		The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.
Line 41:		Line 41:
	The value of <math>\beta\,\!</math> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <math>\beta\,\!</math> is less than, equal to, or greater than one.		The value of <math>\beta\,\!</math> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <math>\beta\,\!</math> is less than, equal to, or greater than one.

	[[Image:WB.8 weibull failure.png\|center\|~~350px~~\| The effect of <math>\beta\,\!</math> on the Weibull failure rate function. ]]		[[Image:WB.8 weibull failure.png\|center\|450px\| The effect of <math>\beta\,\!</math> on the Weibull failure rate function. ]]

	As indicated by above figure, populations with <math>\beta < 1\,\!</math> exhibit a failure rate that decreases with time, populations with <math>\beta = 1\,\!</math> have a constant failure rate (consistent with the exponential distribution) and populations with <math>\beta > 1\,\!</math> have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <math>\beta\,\!</math>. The Weibull failure rate for <math>0 < \beta < 1\,\!</math> is unbounded at <math>T = 0\,\!</math> (or <math>\gamma\,\!)\,\!</math>. The failure rate, <math>\lambda(t),\,\!</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>t\rightarrow \infty\,\!</math> or <math>\lambda (\infty) = 0\,\!</math>. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <math>\beta = 1\,\!</math>, <math>\lambda(t)\,\!</math> yields a constant value of <math> { \frac{1}{\eta }} \,\!</math> or:		As indicated by above figure, populations with <math>\beta < 1\,\!</math> exhibit a failure rate that decreases with time, populations with <math>\beta = 1\,\!</math> have a constant failure rate (consistent with the exponential distribution) and populations with <math>\beta > 1\,\!</math> have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <math>\beta\,\!</math>. The Weibull failure rate for <math>0 < \beta < 1\,\!</math> is unbounded at <math>T = 0\,\!</math> (or <math>\gamma\,\!)\,\!</math>. The failure rate, <math>\lambda(t),\,\!</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>t\rightarrow \infty\,\!</math> or <math>\lambda (\infty) = 0\,\!</math>. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <math>\beta = 1\,\!</math>, <math>\lambda(t)\,\!</math> yields a constant value of <math> { \frac{1}{\eta }} \,\!</math> or:

Chris Kahn: /* Effects of the Shape Parameter, beta */

2013-04-18T21:51:27Z

Effects of the Shape Parameter, beta

← Older revision		Revision as of 21:51, 18 April 2013
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	=== Effects of the Shape Parameter, beta ===		=== Effects of the Shape Parameter, beta ===
	The Weibull shape parameter, <math>\beta\,\!</math>, is also known as the ''slope''. This is because the value of <math>\beta\,\!</math> 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 <math>\beta = 1\,\!</math>, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:		The Weibull shape parameter, <math>\beta\,\!</math>, is also known as the ''slope''. This is because the value of <math>\beta\,\!</math> 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 <math>\beta = 1\,\!</math>, the <i>pdf</i> of the 3-parameter Weibull distribution 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>

Richard House at 18:15, 24 September 2012

2012-09-24T18:15:12Z

Richard House at 20:30, 29 August 2012

2012-08-29T20:30:41Z

Richard House at 20:27, 29 August 2012

2012-08-29T20:27:25Z

← Older revision		Revision as of 20:27, 29 August 2012
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	where <math> \frac{1}{\eta }=\lambda = \,\!</math> failure rate. The parameter <math>\beta\,\!</math> is a pure number, (i.e., it is dimensionless).		where <math> \frac{1}{\eta }=\lambda = \,\!</math> failure rate. The parameter <math>\beta\,\!</math> is a pure number, (i.e., it is dimensionless).
	The following figure shows the effect of different values of the shape parameter, <math>\beta\,\!</math>, on the shape of the ~~<math>~~pdf~~</math>~~. As you can see, the shape can take on a variety of forms based on the value of <math>\beta\,\!</math>.		The following figure shows the effect of different values of the shape parameter, <math>\beta\,\!</math>, on the shape of the ''pdf''. As you can see, the shape can take on a variety of forms based on the value of <math>\beta\,\!</math>.

	[[Image:WB.8 weibull pdf.png\|center\|400px\| The effect of the Weibull shape parameter on the <math>pdf</math>.]]		[[Image:WB.8 weibull pdf.png\|center\|400px\| The effect of the Weibull shape parameter on the <math>pdf</math>.]]

Richard House: /* Effects of the Shape Parameter, β */

2012-08-29T20:24:41Z

Effects of the Shape Parameter, β

← Older revision		Revision as of 20:24, 29 August 2012
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	:*For <math>\beta < 2.6\,\!</math> the Weibull <math>pdf</math> is positively skewed (has a right tail), for <math>2.6 < \beta < 3.7\,\!</math> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal <math>pdf</math> , and for <math>\beta > 3.7\,\!</math> it is negatively skewed (left tail). The way the value of <math>\beta\,\!</math> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <math>\beta = 0.999\,\!</math>, <math>f(0) = \infty\,\!</math>, but for <math>\beta = 1.001\,\!</math>, <math>f(0) = 0.\,\!</math> This abrupt shift is what complicates MLE estimation when <math>\beta\,\!</math> is close to 1.		:*For <math>\beta < 2.6\,\!</math> the Weibull <math>pdf</math> is positively skewed (has a right tail), for <math>2.6 < \beta < 3.7\,\!</math> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal <math>pdf</math> , and for <math>\beta > 3.7\,\!</math> it is negatively skewed (left tail). The way the value of <math>\beta\,\!</math> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <math>\beta = 0.999\,\!</math>, <math>f(0) = \infty\,\!</math>, but for <math>\beta = 1.001\,\!</math>, <math>f(0) = 0.\,\!</math> This abrupt shift is what complicates MLE estimation when <math>\beta\,\!</math> is close to 1.

	'''The Effect of <span class="texhtml">β</span> on the ~~<math>~~cdf~~</math>~~ and Reliability Function'''		'''The Effect of <span class="texhtml">β</span> on the ''cdf'' and Reliability Function'''

	[[Image:WB.8 effect of weibull.png\|center\|400px\| Effect on <math>\beta</math> on the ~~<math>~~cdf~~</math>~~ on the Weibull probability plot with a fixed value of <math>\eta</math> ]]		[[Image:WB.8 effect of weibull.png\|center\|400px\| Effect on <math>\beta</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta</math> ]]

	The above figure shows the effect of the value of <math>\beta\,\!</math> on the ~~<math>~~cdf~~</math>~~, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.		The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.

	[[Image:WB.8 weibull reliability.png\|center\|450px\| The effect of values of <math>\beta</math> on the Weibull reliability plot. ]]		[[Image:WB.8 weibull reliability.png\|center\|450px\| The effect of values of <math>\beta</math> on the Weibull reliability plot. ]]

← Older revision		Revision as of 18:15, 24 September 2012
Line 4:		Line 4:
	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, <math>\beta\,\!</math>, and the scale parameter, <math>\eta\,\!</math>, 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 <math>\gamma = 0,\,\!</math> or the 1-parameter form where <math>\beta = C = \,\!</math> constant, can easily be made.		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, <math>\beta\,\!</math>, and the scale parameter, <math>\eta\,\!</math>, 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 <math>\gamma = 0,\,\!</math> or the 1-parameter form where <math>\beta = C = \,\!</math> constant, can easily be made.

	=== Effects of the Shape Parameter, ~~<span class="texhtml">β</span>~~ ===		=== Effects of the Shape Parameter, beta ===
	The Weibull shape parameter, <math>\beta\,\!</math>, is also known as the ''slope''. This is because the value of <math>\beta\,\!</math> 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 <math>\beta = 1\,\!</math>, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:		The Weibull shape parameter, <math>\beta\,\!</math>, is also known as the ''slope''. This is because the value of <math>\beta\,\!</math> 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 <math>\beta = 1\,\!</math>, 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>

	where <math> \frac{1}{\eta }=\lambda = \,\!</math> failure rate. The parameter <math>\beta\,\!</math> is a pure number, (i.e., it is dimensionless).		where <math> \frac{1}{\eta }=\lambda = \,\!</math> failure rate. The parameter <math>\beta\,\!</math> is a pure number, (i.e., it is dimensionless).
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	For <math> 0<\beta \leq 1 \,\!</math>:		For <math> 0<\beta \leq 1 \,\!</math>:
	:*As <math>t \rightarrow 0\,\!</math> (or <math>\gamma\,\!</math>), <math>f(t)\rightarrow \infty.\,\!</math>		:*As <math>t \rightarrow 0\,\!</math> (or <math>\gamma\,\!</math>), <math>f(t)\rightarrow \infty.\,\!</math>
	:*As <math>t\rightarrow \infty\,\!</math>, <math>f(t)\rightarrow 0</math>.		:*As <math>t\rightarrow \infty\,\!</math>, <math>f(t)\rightarrow 0\,\!</math>.
	:*<math>f(t)\,\!</math> decreases monotonically and is convex as it increases beyond the value of <math>\gamma\,\!</math>.		:*<math>f(t)\,\!</math> decreases monotonically and is convex as it increases beyond the value of <math>\gamma\,\!</math>.
	:*The mode is non-existent.		:*The mode is non-existent.

	For <math> \beta > 1 \,\!</math>:		For <math> \beta > 1 \,\!</math>:
	:*<math>f(t) = 0\,\!</math> at <math>t = 0\,\!</math> (or <math>\gamma\,\!</math>).		:*<math>f(t) = 0\,\!</math> at <math>t = 0\,\!</math> (or <math>\gamma\,\!</math>).
	:*<math>f(t)\,\!</math> increases as <math> t\rightarrow \tilde{T} \,\!</math> (the mode) and decreases thereafter.		:*<math>f(t)\,\!</math> increases as <math> t\rightarrow \tilde{T} \,\!</math> (the mode) and decreases thereafter.
	:*For <math>\beta < 2.6\,\!</math> the Weibull ''pdf'' is positively skewed (has a right tail), for <math>2.6 < \beta < 3.7\,\!</math> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal ''pdf'', and for <math>\beta > 3.7\,\!</math> it is negatively skewed (left tail). The way the value of <math>\beta\,\!</math> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <math>\beta = 0.999\,\!</math>, <math>f(0) = \infty\,\!</math>, but for <math>\beta = 1.001\,\!</math>, <math>f(0) = 0.\,\!</math> This abrupt shift is what complicates MLE estimation when <math>\beta\,\!</math> is close to 1.		:*For <math>\beta < 2.6\,\!</math> the Weibull ''pdf'' is positively skewed (has a right tail), for <math>2.6 < \beta < 3.7\,\!</math> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal ''pdf'', and for <math>\beta > 3.7\,\!</math> it is negatively skewed (left tail). The way the value of <math>\beta\,\!</math> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <math>\beta = 0.999\,\!</math>, <math>f(0) = \infty\,\!</math>, but for <math>\beta = 1.001\,\!</math>, <math>f(0) = 0.\,\!</math> This abrupt shift is what complicates MLE estimation when <math>\beta\,\!</math> is close to 1.

	'''The Effect of ~~<span class="texhtml">β</span>~~ on the ''cdf'' and Reliability Function'''		'''The Effect of beta on the ''cdf'' and Reliability Function'''

	[[Image:WB.8 effect of weibull.png\|center\|400px\| Effect on <math>\beta</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta</math> ]]		[[Image:WB.8 effect of weibull.png\|center\|400px\| Effect on <math>\beta\,\!</math> on the ''cdf'' on the Weibull probability plot with a fixed value of <math>\eta\,\!</math> ]]

	The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.		The above figure shows the effect of the value of <math>\beta\,\!</math> on the ''cdf'', as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of <math>\eta\,\!</math>. The following figure shows the effects of these varied values of <math>\beta\,\!</math> on the reliability plot, which is a linear analog of the probability plot.

	[[Image:WB.8 weibull reliability.png\|center\|450px\| The effect of values of <math>\beta</math> on the Weibull reliability plot. ]]		[[Image:WB.8 weibull reliability.png\|center\|450px\| The effect of values of <math>\beta\,\!</math> on the Weibull reliability plot. ]]

	:*<math>R(t)\,\!</math> decreases sharply and monotonically for <math>0 < \beta < 1\,\!</math> and is convex.		:*<math>R(t)\,\!</math> decreases sharply and monotonically for <math>0 < \beta < 1\,\!</math> and is convex.
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	:*For <math>\beta > 1\,\!</math>, <math>R(t)\,\!</math> decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.		:*For <math>\beta > 1\,\!</math>, <math>R(t)\,\!</math> decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

	'''The Effect of ~~<span class="texhtml">β</span>~~ on the Weibull Failure Rate'''		'''The Effect of beta on the Weibull Failure Rate'''

	The value of <math>\beta\,\!</math> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <math>\beta\,\!</math> is less than, equal to, or greater than one.		The value of <math>\beta\,\!</math> has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of <math>\beta\,\!</math> is less than, equal to, or greater than one.

	[[Image:WB.8 weibull failure.png\|center\|350px\| The effect of <math>\beta</math> on the Weibull failure rate function. ]]		[[Image:WB.8 weibull failure.png\|center\|350px\| The effect of <math>\beta\,\!</math> on the Weibull failure rate function. ]]

	As indicated by above figure, populations with <math>\beta < 1\,\!</math> exhibit a failure rate that decreases with time, populations with <math>\beta = 1\,\!</math> have a constant failure rate (consistent with the exponential distribution) and populations with <math>\beta > 1\,\!</math> have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <math>\beta\,\!</math>. The Weibull failure rate for <math>0 < \beta < 1\,\!</math> is unbounded at <math>T = 0\,\!</math> (or <math>\gamma\,\!)</math>. The failure rate, <math>\lambda(t),\,\!</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>t\rightarrow \infty\,\!</math> or <math>\lambda (\infty) = 0\,\!</math>. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <math>\beta = 1\,\!</math>, <math>\lambda(t)\,\!</math> yields a constant value of <math> { \frac{1}{\eta }} \,\!</math> or:		As indicated by above figure, populations with <math>\beta < 1\,\!</math> exhibit a failure rate that decreases with time, populations with <math>\beta = 1\,\!</math> have a constant failure rate (consistent with the exponential distribution) and populations with <math>\beta > 1\,\!</math> have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of <math>\beta\,\!</math>. The Weibull failure rate for <math>0 < \beta < 1\,\!</math> is unbounded at <math>T = 0\,\!</math> (or <math>\gamma\,\!)\,\!</math>. The failure rate, <math>\lambda(t),\,\!</math> decreases thereafter monotonically and is convex, approaching the value of zero as <math>t\rightarrow \infty\,\!</math> or <math>\lambda (\infty) = 0\,\!</math>. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For <math>\beta = 1\,\!</math>, <math>\lambda(t)\,\!</math> yields a constant value of <math> { \frac{1}{\eta }} \,\!</math> or:

	::<math> \lambda (t)=\lambda ={\frac{1}{\eta }} </math>		::<math> \lambda (t)=\lambda ={\frac{1}{\eta }} \,\!</math>

	This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.		This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.
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	When <math>\beta > 2,\,\!</math> the <math>\lambda(t)\,\!</math> curve is convex, with its slope increasing as <math>t\,\!</math> increases. Consequently, the failure rate increases at an increasing rate as <math>t\,\!</math> increases, indicating wearout life.		When <math>\beta > 2,\,\!</math> the <math>\lambda(t)\,\!</math> curve is convex, with its slope increasing as <math>t\,\!</math> increases. Consequently, the failure rate increases at an increasing rate as <math>t\,\!</math> increases, indicating wearout life.

	=== Effects of the Scale Parameter, ~~<span class="texhtml">η</span>~~ ===		=== Effects of the Scale Parameter, eta ===
	[[Image:WB.8 effects of n.png\|center\|350px\| The effects of <math>\eta</math> on the Weibull ''pdf'' for a common <math>\beta</math>. ]]		[[Image:WB.8 effects of n.png\|center\|350px\| The effects of <math>\eta\,\!</math> on the Weibull ''pdf'' for a common <math>\beta\,\!</math>. ]]

	A change in the scale parameter <math>\eta\,\!</math> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <math>\eta\,\!</math> while holding <math>\beta\,\!</math> constant has the effect of stretching out the ''pdf''. Since the area under a ''pdf'' curve is a constant value of one, the "peak" of the ''pdf'' curve will also decrease with the increase of <math>\eta\,\!</math>, as indicated in the above figure.		A change in the scale parameter <math>\eta\,\!</math> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <math>\eta\,\!</math> while holding <math>\beta\,\!</math> constant has the effect of stretching out the ''pdf''. Since the area under a ''pdf'' curve is a constant value of one, the "peak" of the ''pdf'' curve will also decrease with the increase of <math>\eta\,\!</math>, as indicated in the above figure.
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	:*<math>\eta\,\!</math> has the same units as <math>t\,\!</math>, such as hours, miles, cycles, actuations, etc.		:*<math>\eta\,\!</math> has the same units as <math>t\,\!</math>, such as hours, miles, cycles, actuations, etc.

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

	[[Image:WB.8 location parameter.png\|center\|350px\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull ''pdf''. ]]		[[Image:WB.8 location parameter.png\|center\|350px\| The effect of a positive location parameter, <math>\gamma\,\!</math>, on the position of the Weibull ''pdf''. ]]

	:*When <math>\gamma = 0,\,\!</math> the distribution starts at <math>t=0\,\!</math> or at the origin.		:*When <math>\gamma = 0,\,\!</math> the distribution starts at <math>t=0\,\!</math> or at the origin.

← Older revision		Revision as of 20:30, 29 August 2012
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	The following figure shows the effect of different values of the shape parameter, <math>\beta\,\!</math>, on the shape of the ''pdf''. As you can see, the shape can take on a variety of forms based on the value of <math>\beta\,\!</math>.		The following figure shows the effect of different values of the shape parameter, <math>\beta\,\!</math>, on the shape of the ''pdf''. As you can see, the shape can take on a variety of forms based on the value of <math>\beta\,\!</math>.

	[[Image:WB.8 weibull pdf.png\|center\|400px\| The effect of the Weibull shape parameter on the ~~<math>~~pdf~~</math>~~.]]		[[Image:WB.8 weibull pdf.png\|center\|400px\| The effect of the Weibull shape parameter on the ''pdf''.]]

	For <math> 0<\beta \leq 1 \,\!</math>:		For <math> 0<\beta \leq 1 \,\!</math>:
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	:*<math>f(t) = 0\,\!</math> at <math>t = 0\,\!</math> (or <math>\gamma\,\!</math>).		:*<math>f(t) = 0\,\!</math> at <math>t = 0\,\!</math> (or <math>\gamma\,\!</math>).
	:*<math>f(t)\,\!</math> increases as <math> t\rightarrow \tilde{T} \,\!</math> (the mode) and decreases thereafter.		:*<math>f(t)\,\!</math> increases as <math> t\rightarrow \tilde{T} \,\!</math> (the mode) and decreases thereafter.
	:*For <math>\beta < 2.6\,\!</math> the Weibull ~~<math>~~pdf~~</math>~~ is positively skewed (has a right tail), for <math>2.6 < \beta < 3.7\,\!</math> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal ~~<math>~~pdf~~</math>~~ , and for <math>\beta > 3.7\,\!</math> it is negatively skewed (left tail). The way the value of <math>\beta\,\!</math> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <math>\beta = 0.999\,\!</math>, <math>f(0) = \infty\,\!</math>, but for <math>\beta = 1.001\,\!</math>, <math>f(0) = 0.\,\!</math> This abrupt shift is what complicates MLE estimation when <math>\beta\,\!</math> is close to 1.		:*For <math>\beta < 2.6\,\!</math> the Weibull ''pdf'' is positively skewed (has a right tail), for <math>2.6 < \beta < 3.7\,\!</math> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal ''pdf'', and for <math>\beta > 3.7\,\!</math> it is negatively skewed (left tail). The way the value of <math>\beta\,\!</math> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <math>\beta = 0.999\,\!</math>, <math>f(0) = \infty\,\!</math>, but for <math>\beta = 1.001\,\!</math>, <math>f(0) = 0.\,\!</math> This abrupt shift is what complicates MLE estimation when <math>\beta\,\!</math> is close to 1.

	'''The Effect of <span class="texhtml">β</span> on the ''cdf'' and Reliability Function'''		'''The Effect of <span class="texhtml">β</span> on the ''cdf'' and Reliability Function'''
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	=== Effects of the Scale Parameter, <span class="texhtml">η</span> ===		=== Effects of the Scale Parameter, <span class="texhtml">η</span> ===
	[[Image:WB.8 effects of n.png\|center\|350px\| 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\|350px\| The effects of <math>\eta</math> on the Weibull ''pdf'' 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 ~~<math>~~pdf~~</math>~~. Since the area under a ~~<math>~~pdf~~</math>~~ curve is a constant value of one, the "peak" of the ~~<math>~~pdf~~</math>~~ curve will also decrease with the increase of <~~span class="texhtml"~~>η</~~span~~>, as indicated in the above figure.		A change in the scale parameter <math>\eta\,\!</math> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <math>\eta\,\!</math> while holding <math>\beta\,\!</math> constant has the effect of stretching out the ''pdf''. Since the area under a ''pdf'' curve is a constant value of one, the "peak" of the ''pdf'' curve will also decrease with the increase of <math>\eta\,\!</math>, as indicated in the above figure.

	:*If <math>\eta\,\!</math> is increased while <math>\beta\,\!</math> and <math>\gamma\,\!</math> are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.		:*If <math>\eta\,\!</math> is increased while <math>\beta\,\!</math> and <math>\gamma\,\!</math> are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
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	The location parameter, <math>\gamma\,\!</math>, as the name implies, locates the distribution along the abscissa. Changing the value of <math>\gamma\,\!</math> has the effect of ''sliding'' the distribution and its associated function either to the right (if <math>\gamma > 0\,\!</math>) or to the left (if <math>\gamma < 0\,\!</math>).		The location parameter, <math>\gamma\,\!</math>, as the name implies, locates the distribution along the abscissa. Changing the value of <math>\gamma\,\!</math> has the effect of ''sliding'' the distribution and its associated function either to the right (if <math>\gamma > 0\,\!</math>) or to the left (if <math>\gamma < 0\,\!</math>).

	[[Image:WB.8 location parameter.png\|center\|350px\| 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\|350px\| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull ''pdf''. ]]

	:*When <math>\gamma = 0,\,\!</math> the distribution starts at <math>t=0\,\!</math> or at the origin.		:*When <math>\gamma = 0,\,\!</math> the distribution starts at <math>t=0\,\!</math> or at the origin.