Example: The Effect of Beta on the Weibull pdf: Difference between revisions

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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>.
 
[[Image:WB.8 weibull pdf.png|center|400px| The effect of the Weibull shape parameter on the <math>pdf</math>.]]
 
 
For <math> 0<\beta \leq 1 </math>:
:*As <span class="texhtml">''t''→0</span> <span class="texhtml">(</span>or <span class="texhtml">γ),</span> <span class="texhtml">''f''(''t'')→∞.</span>
:*As <span class="texhtml">''t''→∞</span>, <span class="texhtml">''f''(''t'')→0</span>.
:*<span class="texhtml">''f''(''t'')</span> decreases monotonically and is convex as  increases beyond the value of <span class="texhtml">γ</span>.
:*The mode is non-existent.
 
For <span class="texhtml">β &gt; 1</span>:
 
:*<span class="texhtml">''f''(''t'') = 0</span> at  <span class="texhtml">(</span>or <span class="texhtml">γ)</span>.
:*<span class="texhtml">''f''(''t'')</span> increases as <math> t\rightarrow \tilde{T} </math> (the mode) and decreases thereafter.
:*For <span class="texhtml">β &lt; 2.6</span> the Weibull <math>pdf</math> 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 <math>pdf</math> , and for <span class="texhtml">β &gt; 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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