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| | #REDIRECT [[The Weibull Distribution]] |
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| | valign="middle" align="left" bgcolor=EEEDF7|[[Image: Weibull-Examples-banner.png|400px|center]]
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| <br>
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| [[Example: The Effect of Beta on the Weibull pdf]]
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| The following Figure shows the effect of different values of the shape parameter, <span class="texhtml">β</span>, on the shape of the <math>pdf</math>. One can see that the shape of the can take on a variety of forms based on the value of <span class="texhtml">β</span>.
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| [[Image:WB.8 weibull pdf.png|center|400px| The effect of the Weibull shape parameter on the <math>pdf</math>.]]
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| For <math> 0<\beta \leq 1 </math>:
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| :*As <span class="texhtml">''t''→0</span> <span class="texhtml">(</span>or <span class="texhtml">γ),</span> <span class="texhtml">''f''(''t'')→∞.</span>
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| :*As <span class="texhtml">''t''→∞</span>, <span class="texhtml">''f''(''t'')→0</span>.
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| :*<span class="texhtml">''f''(''t'')</span> decreases monotonically and is convex as increases beyond the value of <span class="texhtml">γ</span>.
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| :*The mode is non-existent.
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| For <span class="texhtml">β > 1</span>:
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| :*<span class="texhtml">''f''(''t'') = 0</span> at <span class="texhtml">(</span>or <span class="texhtml">γ)</span>.
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| :*<span class="texhtml">''f''(''t'')</span> increases as <math> t\rightarrow \tilde{T} </math> (the mode) and decreases thereafter.
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| :*For <span class="texhtml">β < 2.6</span> the Weibull <math>pdf</math> 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 <math>pdf</math> , and for <span class="texhtml">β > 3.7</span> it is negatively skewed (left tail). The way the value of <span class="texhtml">β</span> relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for <span class="texhtml">β = 0.999</span>, <span class="texhtml">''f''(0) = ∞</span>, but for <span class="texhtml">β = 1.001</span>, <span class="texhtml">''f''(0) = 0.</span> This abrupt shift is what complicates MLE estimation when <span class="texhtml">β</span> is close to one.
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