Template:The effect of beta on Weibull failure rate - Revision history

Lisa Hacker: Redirected page to Weibull Distribution Characteristics

2012-08-07T02:53:49Z

Redirected page to Weibull Distribution Characteristics

← Older revision		Revision as of 02:53, 7 August 2012
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	~~'''The Effect of <span class="texhtml">β</span> on the Weibull Failure Rate'''~~		#REDIRECT [[Weibull Distribution Characteristics]]

	The value of <span class="texhtml">β</span> 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 <span class="texhtml">β</span> is less than, equal to, or greater than one.


	[[~~Image:WB.8 weibull failure.png\|center\|250px\| The effect of <math>\beta</math> on the~~ Weibull ~~failure rate function.~~ ]]


	As indicated by above Figure, populations with <span class="texhtml">β < 1</span> exhibit a failure rate that decreases with time, populations with <span class="texhtml">β = 1</span> have a constant failure rate (consistent with the exponential distribution) and populations with <span class="texhtml">β > 1</span> 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 <span class="texhtml">β.</span> The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">(</span>or <span class="texhtml">γ)</span>. The failure rate, <span class="texhtml">λ(''t''),</span> decreases thereafter monotonically and is convex, approaching the value of zero as <span class="texhtml">''t''→∞</span> or <span class="texhtml">λ(∞) = 0</span>. 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 <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''t'')</span> yields a constant value of <math> { \frac{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 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 increases.

	For <span class="texhtml">β = 2</span> there emerges a straight line relationship between <span class="texhtml">λ(''t'')</span> and , starting at a value of <span class="texhtml">λ(''t'') = 0</span> at <span class="texhtml">''t'' = γ</span>, and increasing thereafter with a slope of <math> { \frac{2}{\eta ^{2}}} </math>. Consequently, the failure rate increases at a constant rate as increases. Furthermore, if <span class="texhtml">η = 1</span> the slope becomes equal to 2, and when <span class="texhtml">γ = 0</span>, <span class="texhtml">λ(''t'')</span> becomes a straight line which passes through the origin with a slope of 2. Note that at <span class="texhtml">β = 2</span>, the Weibull distribution equations reduce to that of the Rayleigh distribution.

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

	~~<br>~~

Nicolette Young at 18:01, 25 April 2012

2012-04-25T18:01:35Z

← Older revision		Revision as of 18:01, 25 April 2012
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	[[Image:WB.8 weibull failure.png\|center\|~~400px~~\| The effect of <math>\beta</math> on the Weibull failure rate function. ]]		[[Image:WB.8 weibull failure.png\|center\|250px\| The effect of <math>\beta</math> on the Weibull failure rate function. ]]

Nicolette Young at 16:38, 15 March 2012

2012-03-15T16:38:10Z

← Older revision		Revision as of 16:38, 15 March 2012
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	[[Image:~~lda6~~.4.~~gif\|thumb~~\|center\|400px\| The effect of <math>\beta</math> on the Weibull failure rate function. ]]		[[Image:WB.8 weibull failure.png\|center\|400px\| The effect of <math>\beta</math> on the Weibull failure rate function. ]]

Harry Guo at 21:57, 9 February 2012

2012-02-09T21:57:52Z

← Older revision		Revision as of 21:57, 9 February 2012
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	'''The Effect of <span class="texhtml">β</span> on the Weibull Failure Rate'''		'''The Effect of <span class="texhtml">β</span> on the Weibull Failure Rate'''

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	As indicated by Figure ~~6-4~~, populations with <span class="texhtml">β < 1</span> exhibit a failure rate that decreases with time, populations with <span class="texhtml">β = 1</span> have a constant failure rate (consistent with the exponential distribution) and populations with <span class="texhtml">β > 1</span> 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 <span class="texhtml">β.</span> The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">(</span>or <span class="texhtml">γ)</span>. The failure rate, <span class="texhtml">λ(''t''),</span> decreases thereafter monotonically and is convex, approaching the value of zero as <span class="texhtml">''t''→∞</span> or <span class="texhtml">λ(∞) = 0</span>. 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 <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''t'')</span> yields a constant value of <math> { \frac{1}{\eta }} </math> or:		As indicated by above Figure, populations with <span class="texhtml">β < 1</span> exhibit a failure rate that decreases with time, populations with <span class="texhtml">β = 1</span> have a constant failure rate (consistent with the exponential distribution) and populations with <span class="texhtml">β > 1</span> 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 <span class="texhtml">β.</span> The Weibull failure rate for <span class="texhtml">0 < β < 1</span> is unbounded at <span class="texhtml">(</span>or <span class="texhtml">γ)</span>. The failure rate, <span class="texhtml">λ(''t''),</span> decreases thereafter monotonically and is convex, approaching the value of zero as <span class="texhtml">''t''→∞</span> or <span class="texhtml">λ(∞) = 0</span>. 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 <span class="texhtml">β = 1</span>, <span class="texhtml">λ(''t'')</span> 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>

Harry Guo: Created page with ' '''The Effect of $β$ on the Weibull Failure Rate''' The value of $β$ has a marked effect on the failure rate of the Weib…'

2012-02-09T21:51:26Z

Created page with ' '''The Effect of β on the Weibull Failure Rate''' The value of β has a marked effect on the failure rate of the Weib…'

New page

'''The Effect of β on the Weibull Failure Rate'''

The value of β 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 β is less than, equal to, or greater than one.

[[Image:lda6.4.gif|thumb|center|400px| The effect of <math>\beta</math> on the Weibull failure rate function. ]]

As indicated by Figure 6-4, populations with β < 1 exhibit a failure rate that decreases with time, populations with β = 1 have a constant failure rate (consistent with the exponential distribution) and populations with β > 1 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 β. The Weibull failure rate for 0 < β < 1 is unbounded at (or γ). The failure rate, λ(''t''), decreases thereafter monotonically and is convex, approaching the value of zero as ''t''→∞ or λ(∞) = 0. 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 β = 1, λ(''t'') yields a constant value of <math> { \frac{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 β > 1, λ(''t'') increases as increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For 1 < β < 2, the λ(''t'') curve is concave, consequently the failure rate increases at a decreasing rate as increases.

For β = 2 there emerges a straight line relationship between λ(''t'') and , starting at a value of λ(''t'') = 0 at ''t'' = γ, and increasing thereafter with a slope of <math> { \frac{2}{\eta ^{2}}} </math>. Consequently, the failure rate increases at a constant rate as increases. Furthermore, if η = 1 the slope becomes equal to 2, and when γ = 0, λ(''t'') becomes a straight line which passes through the origin with a slope of 2. Note that at β = 2, the Weibull distribution equations reduce to that of the Rayleigh distribution.

When β > 2, the λ(''t'') curve is convex, with its slope increasing as increases. Consequently, the failure rate increases at an increasing rate as increases indicating wear-out life.

Template:The effect of beta on Weibull failure rate - Revision history

Lisa Hacker: Redirected page to Weibull Distribution Characteristics

Nicolette Young at 18:01, 25 April 2012

Nicolette Young at 16:38, 15 March 2012

Harry Guo at 21:57, 9 February 2012

Harry Guo: Created page with ' '''The Effect of β on the Weibull Failure Rate''' The value of β has a marked effect on the failure rate of the Weib…'

Harry Guo: Created page with ' '''The Effect of $β$ on the Weibull Failure Rate''' The value of $β$ has a marked effect on the failure rate of the Weib…'