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===Characteristics===
#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]]
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">''p''''d''''f''</span> as shown in the next figure. In addition, when the <span class="texhtml">''c''''d''''f''</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.gif|thumb|center|400px]] <br>
 
:*For <span class="texhtml">0 &lt; β &lt; 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 &gt; 1</span> , <span class="texhtml">''f''(''T''),</span> the Weibull 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 &lt; 2.6</span> the Weibull <span class="texhtml">''p''''d''''f''</span> is positively skewed (has a right tail), for <span class="texhtml">2.6 &lt; β &lt; 3.7</span> its coefficient of skewness approaches zero (no tail); consequently, it may approximate the normal <span class="texhtml">''p''''d''''f''</span> , and for <span class="texhtml">β &gt; 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.gif|thumb|center|400px]]
 
<br> [[Image:ALTA4.5.gif|thumb|center|400px]]
 
<br>
 
:*<span class="texhtml">''R''(''T'')</span> decreases sharply and monotonically for <span class="texhtml">0 &lt; β &lt; 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 &lt; β &lt; 1</span> , and is convex.
:*For <span class="texhtml">β &gt; 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.
 
<br>
 
'''Effects of <span class="texhtml">β</span> on the Failure Rate Function'''
 
 
[[Image:ALTA4.6.gif|thumb|center|400px]] <br>
 
:*The Weibull failure rate for <span class="texhtml">0 &lt; β &lt; 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.
::*Inadequate quality control.
::*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 &gt; 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 &lt; β &lt; 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 &gt; 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  <math>\eta </math>====
Eta,  <math>\eta ,</math>  is called the scale parameter of the Weibull distribution. The parameter  <math>\eta </math>  has the same units as  <math>T</math> , such as hours, miles, cycles, actuations, etc.
<br>
[[Image:ALTA4.7.gif|thumb|center|300px| ]]
<br>
:• A change in the scale parameter  <math>\eta </math>  has the same effect on the distribution as a change of the abscissa scale.
::o If  <math>\eta </math>  is increased while  <math>\beta </math>  is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
::o If  <math>\eta </math>  is decreased while  <math>\beta </math>  is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.
<br>

Latest revision as of 01:56, 16 August 2012

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