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== Characteristics of the Weibull Distribution ==
#REDIRECT [[Weibull Distribution Characteristics]]
 
As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, <span class="texhtml">β</span>, and the scale parameter, <span class="texhtml">η</span>, affect such distribution characteristics as the shape of the  curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where <span class="texhtml">γ = 0,</span> or the 1-parameter form where <span class="texhtml">β = ''C'' = </span>constant, can easily be made.
 
=== Characteristic Effects of the Shape Parameter, <span class="texhtml">β</span>  ===
 
The Weibull shape parameter, <span class="texhtml">β</span>, is also known as the ''slope''. This is because the value of <span class="texhtml">β</span> is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when <span class="texhtml">β = 1</span>, the  of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:
 
::<math> f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} </math>
 
where <math> \frac{1}{\eta }=\lambda = </math> failure rate. The parameter <span class="texhtml">β</span> is a pure number, i.e. it is dimensionless.
 
 
{{The effect of beta on the Weibull pdf}}
 
 
{{effect of beta on the cdf}}
 
{{The effect of beta on Weibull failure rate}}
 
=== Characteristic Effects of the Scale Parameter, <span class="texhtml">η</span> ===
 
[[Image:WB.8 effects of n.png|center|250px| The effects of <math>\eta</math> on the Weibull <math>pdf</math> for a common <math>\beta</math>. ]]
 
A change in the scale parameter <span class="texhtml">η</span> has the same effect on the distribution as a change of the abscissa scale. Increasing the value of <span class="texhtml">η</span> while holding <span class="texhtml">β</span> constant has the effect of stretching out the . Since the area under a  curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of <span class="texhtml">η</span>, as indicated in the above figure.
 
:*If <span class="texhtml">η</span> is increased while <span class="texhtml">β</span> and <span class="texhtml">γ</span> are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
:*If <span class="texhtml">η</span> is decreased while <span class="texhtml">β</span> and <span class="texhtml">γ</span> are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or <span class="texhtml">γ</span>), and its height increases.
:*<span class="texhtml">η</span> has the same units as , such as hours, miles, cycles, actuations, etc.
 
<br>
 
=== Characteristic Effects of the Location Parameter, <span class="texhtml">γ</span> ===
 
The location parameter, <span class="texhtml">γ</span>, as the name implies, locates the distribution along the abscissa. Changing the value of <span class="texhtml">γ</span> has the effect of ''sliding'' the distribution and its associated function either to the right (if <span class="texhtml">γ &gt; 0</span>) or to the left (if <span class="texhtml">γ &lt; 0</span>).''
 
 
[[Image:WB.8 location parameter.png|center|400px| The effect of a positive location parameter, <math>\gamma</math>, on the position of the Weibull <math>pdf</math>. ]]
 
:*When <span class="texhtml">γ = 0,</span> the distribution starts at  or at the origin.
:*If <span class="texhtml">γ &gt; 0,</span> the distribution starts at the location <span class="texhtml">γ</span> to the right of the origin.
:*If <span class="texhtml">γ &lt; 0,</span> the distribution starts at the location <span class="texhtml">γ</span> to the left of the origin.
:*<span class="texhtml">γ</span> provides an estimate of the earliest time-to-failure of such units.
:*The life period 0 to <span class="texhtml">+ γ</span> is a failure free operating period of such units.
:*The parameter <span class="texhtml">γ</span> may assume all values and provides an estimate of the earliest time a failure may be observed. A negative <span class="texhtml">γ</span> may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
:*<span class="texhtml">γ</span> has the same units as T, such as hours, miles, cycles, actuations, etc.

Latest revision as of 02:50, 7 August 2012

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