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Richard House: Redirected page to Distributions Used in Accelerated Testing#The Weibull Distribution

2012-08-16T01:56:18Z

Redirected page to Distributions Used in Accelerated Testing#The Weibull Distribution

← Older revision		Revision as of 01:56, 16 August 2012
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	#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]		#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]]

Lisa Hacker: Redirected page to Distributions Used in Accelerated Testing

2012-08-09T10:15:23Z

Redirected page to Distributions Used in Accelerated Testing

← Older revision		Revision as of 10:15, 9 August 2012
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	~~===Characteristics===~~		#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]
	The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, <math>\beta ,</math> and eta, <math>\eta ,</math> and the effect they have on the <math>pdf,</math> reliability and failure rate functions.

	~~==== Looking at <span class="texhtml">β</span> ====~~

	Beta (<span class="texhtml">β</span>) is called the shape parameter or slope of the Weibull distribution. Changing the value of <span class="texhtml">β</span> forces a change in the shape of the <span class="texhtml">''pdf''</span> as shown in the next figure. In addition, when the <span class="texhtml">''cdf''</span> is plotted on Weibull probability paper, a change in beta is a change in the slope of the distribution on Weibull probability paper.

	~~<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''~~

	~~<br>~~ [[~~Image:ALTA4.3.png\|center\|400px~~]] ~~<br>~~

	~~:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:~~
	~~::*As <math>T\to 0,</math> <math>f(T)\to \infty</math>.~~
	~~::*As <math>T\to \infty </math> , <math>f(T)\to 0</math>.~~
	~~::*<span class="texhtml">''f''(''T'')</span> decreases monotonically and is convex as <span class="texhtml">''T''</span> increases.~~
	~~::*The mode is non-existent.~~
	~~:*For <span class="texhtml">β = 1,</span> it becomes the exponential distribution, as a special case, <br>~~

	~~<br>~~

	~~:or:~~

	~~<br>~~

	~~::<math>f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta >0,T\ge 0</math>~~

	~~<br>~~

	~~:where <math>\tfrac{1}{\eta }=\lambda =</math> chance, useful life, or failure rate.~~

	~~<br>~~

	~~:*For <span class="texhtml">β > 1</span>,  <span class="texhtml">''f''(''T'')</span>, the Weibull distribution assumes wear-out type shapes (i.e., the failure rate increases with time) and:~~
	~~::*<span class="texhtml">''f''(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> .~~
	~~::*<span class="texhtml">''f''(''T'')</span> increases as <math>T\to \tilde{T}</math> (mode) and decreases thereafter.~~
	::*For <span class="texhtml">β = 2</span> it becomes the Rayleigh distribution as a special case. For <span class="texhtml">β < 2.6</span>, the Weibull <span class="texhtml">''pdf''</span> is positively skewed (has a right tail). For <span class="texhtml">2.6 < β < 3.7</span>, its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal <span class="texhtml">''pdf''</span>, and for <span class="texhtml">β > 3.7</span> it is negatively skewed (left tail).
	~~:*The parameter <span class="texhtml">β</span> is a pure number (i.e., it is dimensionless).~~

	~~<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''~~

	~~<br> [[Image:ALTA4.4.png\|center\|400px]]~~

	~~<br> [[Image:ALTA4.5.png\|center\|400px]]~~

	~~<br>~~

	:*<span class="texhtml">''R''(''T'')</span> decreases sharply and monotonically for <span class="texhtml">0 < β < 1</span>. It is convex and decreases less sharply for the same <span class="texhtml">β</span>.
	:*For <span class="texhtml">β = 1</span> and the same <span class="texhtml">η</span>, <span class="texhtml">''R''(''T'')</span> decreases monotonically but less sharply than for <span class="texhtml">0 < β < 1</span>, and it is convex.
	:*For <span class="texhtml">β > 1</span>, <span class="texhtml">''R''(''T'')</span> decreases as <span class="texhtml">''T''</span> increases but less sharply than before. As wear-out sets in, it decreases sharply and goes through an inflection point.

	~~<br>~~

	~~'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''~~

	~~<br> [[Image:ALTA4.6.png\|center\|400px]] <br>~~

	:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:
	~~::*Burn-in testing and/or environmental stress screening are not well implemented.~~
	~~::*There are problems in the production line.~~
	~~::*There is inadequate quality control.~~
	~~::*There are packaging and transit problems.~~
	~~:*For <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''T'')</span> yields a constant value of <math>\tfrac{1}{\eta }</math> , or:~~
	~~::<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.~~

	:*For <span class="texhtml">β > 1</span>, <span class="texhtml">λ(''T'')</span> increases as <span class="texhtml">''T''</span> increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For <span class="texhtml">1 < β < 2</span>, the <span class="texhtml">λ(''T'')</span> curve is concave. Consequently, the failure rate increases at a decreasing rate as <span class="texhtml">''T''</span> increases.
	~~:*For <span class="texhtml">β = 2</span>, or for the Rayleigh distribution case, the failure rate function is given by:~~
	~~::<math>\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)</math>~~

	<br> Hence there emerges a straight line relationship between <span class="texhtml">λ(''T'')</span> and <span class="texhtml">''T''</span>, starting at a value of <span class="texhtml">λ(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> and increasing thereafter with a slope of <math>\tfrac{2}{{{\eta }^{2}}}</math> . Consequently, the failure rate increases at a constant rate as <span class="texhtml">''T''</span> increases. Furthermore, if <span class="texhtml">η = 1</span> the slope becomes equal to 2, and <span class="texhtml">λ(''T'')</span> becomes a straight line which passes through the origin with a slope of 2.

	:*When <span class="texhtml">β > 2</span> the <span class="texhtml">λ(''T'')</span> curve is convex, with its slope increasing as <span class="texhtml">''T''</span> increases. Consequently, the failure rate increases at an increasing rate as <span class="texhtml">''T''</span> increases, indicating wear-out life.

	~~<br>~~

	~~==== Looking at <span class="texhtml">η</span> ====~~

	Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.

	~~<br> [[Image:ALTA4.7.png\|center\|400px]] <br>~~

	~~:*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.~~
	::*If <span class="texhtml">η</span> is increased while <span class="texhtml">β</span> is 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> is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.

	~~<br>~~

Chris Kahn: /* Looking at η */

2012-06-25T10:08:01Z

Looking at η

← Older revision		Revision as of 10:08, 25 June 2012
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	Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.		Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.

	<br> [[Image:ALTA4.7.png\|center\|~~200px~~]] <br>		<br> [[Image:ALTA4.7.png\|center\|400px]] <br>

	:*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.		:*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.

Chris Kahn: /* Looking at β */

2012-06-25T10:07:40Z

Looking at β

← Older revision		Revision as of 10:07, 25 June 2012
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	<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''		<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''

	<br> [[Image:ALTA4.3.png\|center\|~~200px~~]] <br>		<br> [[Image:ALTA4.3.png\|center\|400px]] <br>

	:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:		:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:
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	<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''		<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''

	<br> [[Image:ALTA4.4.png\|center\|~~200px~~]]		<br> [[Image:ALTA4.4.png\|center\|400px]]

	<br> [[Image:ALTA4.5.png\|center\|~~200px~~]]		<br> [[Image:ALTA4.5.png\|center\|400px]]

	<br>		<br>
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	'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''		'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''

	<br> [[Image:ALTA4.6.png\|center\|~~200px~~]] <br>		<br> [[Image:ALTA4.6.png\|center\|400px]] <br>

	:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:		:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:

Chuck Smith: /* Looking at η */

2012-04-26T16:05:44Z

Looking at η

← Older revision		Revision as of 16:05, 26 April 2012
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	Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.		Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.

	<br> [[Image:ALTA4.7.png\|center\|~~400px~~]] <br>		<br> [[Image:ALTA4.7.png\|center\|200px]] <br>

	:*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.		:*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.

Chuck Smith: /* Looking at β */

2012-04-26T16:05:23Z

Looking at β

← Older revision		Revision as of 16:05, 26 April 2012
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	<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''		<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''

	<br> [[Image:ALTA4.3.png\|center\|~~400px~~]] <br>		<br> [[Image:ALTA4.3.png\|center\|200px]] <br>

	:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:		:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:
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	<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''		<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''

	<br> [[Image:ALTA4.4.png\|center\|~~400px~~]]		<br> [[Image:ALTA4.4.png\|center\|200px]]

	<br> [[Image:ALTA4.5.png\|center\|~~400px~~]]		<br> [[Image:ALTA4.5.png\|center\|200px]]

	<br>		<br>
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	'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''		'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''

	<br> [[Image:ALTA4.6.png\|center\|~~400px~~]] <br>		<br> [[Image:ALTA4.6.png\|center\|200px]] <br>

	:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:		:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:

Nicolette Young: /* Looking at η */

2012-03-20T16:50:19Z

Looking at η

← Older revision		Revision as of 16:50, 20 March 2012
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	Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.		Eta (<span class="texhtml">η</span>) is called the scale parameter of the Weibull distribution. The parameter <span class="texhtml">η</span> has the same units as <span class="texhtml">''T''</span>, such as hours, miles, cycles, actuations, etc.

	<br> [[Image:ALTA4.7.~~gif\|thumb~~\|center\|~~300px~~]] <br>		<br> [[Image:ALTA4.7.png\|center\|400px]] <br>

	:*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.		:*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.

Nicolette Young: /* Looking at β */

2012-03-20T16:47:48Z

Looking at β

← Older revision		Revision as of 16:47, 20 March 2012
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	<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''		<br> '''Effects of <span class="texhtml">β</span> on the ''pdf'''''

	<br> [[Image:ALTA4.3.~~gif\|thumb~~\|center\|400px]] <br>		<br> [[Image:ALTA4.3.png\|center\|400px]] <br>

	:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:		:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:
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	<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''		<br> '''Effects of <span class="texhtml">β</span> on the Reliability Function and the ''cdf'''''

	<br> [[Image:ALTA4.4.~~gif\|thumb~~\|center\|400px]]		<br> [[Image:ALTA4.4.png\|center\|400px]]

	<br> [[Image:ALTA4.5.~~gif\|thumb~~\|center\|400px]]		<br> [[Image:ALTA4.5.png\|center\|400px]]

	<br>		<br>
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	'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''		'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''

	<br> [[Image:ALTA4.6.~~gif\|thumb~~\|center\|400px]] <br>		<br> [[Image:ALTA4.6.png\|center\|400px]] <br>

	:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:		:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:

Chris Kahn at 23:51, 6 March 2012

2012-03-06T23:51:24Z

← Older revision		Revision as of 23:51, 6 March 2012
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	:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:		:*For <span class="texhtml">0 < β < 1</span> , the failure rate decreases with time and:
	::*As <math>T\to 0,</math> <math>f(T)\to \infty .</math>		::*As <math>T\to 0,</math> <math>f(T)\to \infty</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>.
	::*<span class="texhtml">''f''(''T'')</span> decreases monotonically and is convex as <span class="texhtml">''T''</span> increases.		::*<span class="texhtml">''f''(''T'')</span> decreases monotonically and is convex as <span class="texhtml">''T''</span> increases.
	::*The mode is non-existent.		::*The mode is non-existent.
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	<br> [[Image:ALTA4.6.gif\|thumb\|center\|400px]] <br>		<br> [[Image:ALTA4.6.gif\|thumb\|center\|400px]] <br>

	:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:		:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:
	::*Burn-in testing and/or environmental stress screening are not well implemented.		::*Burn-in testing and/or environmental stress screening are not well implemented.
	::*There are problems in the production line.		::*There are problems in the production line.

Chris Kahn at 23:49, 6 March 2012

2012-03-06T23:49:34Z

← Older revision		Revision as of 23:49, 6 March 2012
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	<br>		<br>

	:*For <span class="texhtml">β > 1</span> , <span class="texhtml">''f''(''T''),</span> the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:		:*For <span class="texhtml">β > 1</span>,  <span class="texhtml">''f''(''T'')</span>, the Weibull distribution assumes wear-out type shapes (i.e., the failure rate increases with time) and:
	::*<span class="texhtml">''f''(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> .		::*<span class="texhtml">''f''(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> .
	::*<span class="texhtml">''f''(''T'')</span> increases as <math>T\to \tilde{T}</math> (mode) and decreases thereafter.		::*<span class="texhtml">''f''(''T'')</span> increases as <math>T\to \tilde{T}</math> (mode) and decreases thereafter.
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	<br>		<br>

	:*<span class="texhtml">''R''(''T'')</span> decreases sharply and monotonically for <span class="texhtml">0 < β < 1</span> ~~, it~~ is convex, and decreases less sharply for the same <span class="texhtml">β</span> .		:*<span class="texhtml">''R''(''T'')</span> decreases sharply and monotonically for <span class="texhtml">0 < β < 1</span>. It is convex and decreases less sharply for the same <span class="texhtml">β</span>.
	:*For <span class="texhtml">β = 1</span> and the same <span class="texhtml">η</span> , <span class="texhtml">''R''(''T'')</span> decreases monotonically but less sharply than for <span class="texhtml">0 < β < 1</span> , and is convex.		:*For <span class="texhtml">β = 1</span> and the same <span class="texhtml">η</span>, <span class="texhtml">''R''(''T'')</span> decreases monotonically but less sharply than for <span class="texhtml">0 < β < 1</span>, and it is convex.
	:*For <span class="texhtml">β > 1</span> , <span class="texhtml">''R''(''T'')</span> decreases as <span class="texhtml">''T''</span> increases but less sharply than before~~, and as~~ wear-out sets in, it decreases sharply and goes through an inflection point.		:*For <span class="texhtml">β > 1</span>, <span class="texhtml">''R''(''T'')</span> decreases as <span class="texhtml">''T''</span> increases but less sharply than before. As wear-out sets in, it decreases sharply and goes through an inflection point.

	<br>		<br>
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	<br> [[Image:ALTA4.6.gif\|thumb\|center\|400px]] <br>		<br> [[Image:ALTA4.6.gif\|thumb\|center\|400px]] <br>

	:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span> . The failure rate, <span class="texhtml">λ(''T''),</span> decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:		:*The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">''T'' = 0</span>. The failure rate, <span class="texhtml">λ(''T'')</span>, decreases thereafter monotonically and is convex, approaching the value of zero as <math>T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:
	::*Burn-in testing and/or environmental stress screening are not well implemented.		::*Burn-in testing and/or environmental stress screening are not well implemented.
	::*There are problems in the production line.		::*There are problems in the production line.
	::*~~Inadequate~~ quality control.		::*There is inadequate quality control.
	::*~~Packaging~~ and transit problems.		::*There are packaging and transit problems.
	:*For <span class="texhtml">β = 1</span> , <span class="texhtml">λ(''T'')</span> yields a constant value of <math>\tfrac{1}{\eta }</math> , or:		:*For <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''T'')</span> yields a constant value of <math>\tfrac{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.

	:*For <span class="texhtml">β > 1</span> , <span class="texhtml">λ(''T'')</span> increases as <span class="texhtml">''T''</span> increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For <span class="texhtml">1 < β < 2</span> the <span class="texhtml">λ(''T'')</span> curve is concave, ~~consequently~~ the failure rate increases at a decreasing rate as <span class="texhtml">''T''</span> increases.		:*For <span class="texhtml">β > 1</span>, <span class="texhtml">λ(''T'')</span> increases as <span class="texhtml">''T''</span> increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For <span class="texhtml">1 < β < 2</span>, the <span class="texhtml">λ(''T'')</span> curve is concave. Consequently, the failure rate increases at a decreasing rate as <span class="texhtml">''T''</span> increases.
	:*For <span class="texhtml">β = 2</span> , or for the Rayleigh distribution case, the failure rate function is given by:		:*For <span class="texhtml">β = 2</span>, or for the Rayleigh distribution case, the failure rate function is given by:
	::<math>\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)</math>		::<math>\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)</math>

	<br> Hence there emerges a straight line relationship between <span class="texhtml">λ(''T'')</span> and <span class="texhtml">''T''</span> , starting at a value of <span class="texhtml">λ(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> and increasing thereafter with a slope of <math>\tfrac{2}{{{\eta }^{2}}}</math> . Consequently, the failure rate increases at a constant rate as <span class="texhtml">''T''</span> increases. Furthermore, if <span class="texhtml">η = 1</span> the slope becomes equal to 2, and <span class="texhtml">λ(''T'')</span> becomes a straight line which passes through the origin with a slope of 2.		<br> Hence there emerges a straight line relationship between <span class="texhtml">λ(''T'')</span> and <span class="texhtml">''T''</span>, starting at a value of <span class="texhtml">λ(''T'') = 0</span> at <span class="texhtml">''T'' = 0</span> and increasing thereafter with a slope of <math>\tfrac{2}{{{\eta }^{2}}}</math> . Consequently, the failure rate increases at a constant rate as <span class="texhtml">''T''</span> increases. Furthermore, if <span class="texhtml">η = 1</span> the slope becomes equal to 2, and <span class="texhtml">λ(''T'')</span> becomes a straight line which passes through the origin with a slope of 2.

	:*When <span class="texhtml">β > 2</span> the <span class="texhtml">λ(''T'')</span> curve is convex, with its slope increasing as <span class="texhtml">''T''</span> increases. Consequently, the failure rate increases at an increasing rate as <span class="texhtml">''T''</span> increases, indicating wear-out life.		:*When <span class="texhtml">β > 2</span> the <span class="texhtml">λ(''T'')</span> curve is convex, with its slope increasing as <span class="texhtml">''T''</span> increases. Consequently, the failure rate increases at an increasing rate as <span class="texhtml">''T''</span> increases, indicating wear-out life.