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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 three-parameter form. The appropriate substitutions to obtain the other forms, such as the two-parameter form where <span class="texhtml">γ = 0,</span> or the one-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 three-parameter Weibull reduces to that of the two-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 weibull on the pdf}}
 
==== The Effect of <span class="texhtml">β</span> on the <math>cdf</math> and Reliability Function ====
 
[[Image:lda6.2.gif|thumb|center|350px| Effect on <math>\beta</math> on the <math>cdf</math> on the Weibull probability plot with a fixed value of <math>\eta</math> ]]
Figure 6-2 shows the effect of the value of <span class="texhtml">β</span> 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 <span class="texhtml">η</span>. Figure 6-3 shows the effects of these varied values of <span class="texhtml">β</span> on the reliability plot, which is a linear analog of the probability plot.
 
[[Image:lda6.3.gif|thumb|center|350px| The effect of values of <math>\beta</math> on the Weibull reliability plot. ]]
<br>
 
:*<span class="texhtml">''R''(''T'')</span> decreases sharply and monotonically for <span class="texhtml">0 &lt; β &lt; 1</span> and is convex.
:*For <span class="texhtml">β = 1</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  increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.
 
<br>
 
==== The Effect of <span class="texhtml">β</span> on the Weibull Failure Rate  ====
 
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: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 <span class="texhtml">β &lt; 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 &gt; 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 &lt; β &lt; 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 &gt; 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 &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  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 &gt; 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>
 
=== Characteristic Effects of the Scale Parameter, <span class="texhtml">η</span> ===
 
 
[[Image:lda6.5.gif|thumb|center|300px| 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 Figure 6-5.
 
:*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:lda6.6.gif|thumb|center|300px| 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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