Template:The effect of beta on the Weibull pdf: Difference between revisions

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'''The Effect of <span class="texhtml">β</span> on the <math>pdf</math>'''
'''The Effect of <span class="texhtml">β</span> on the <math>pdf</math>'''


Figure 6-1 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>.  
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:lda6.1.gif|thumb|center|350px| Figure 6-1: The effect of the Weibull shape parameter on the <math>pdf</math>.]]
[[Image:lda6.1.gif|thumb|center|350px| The effect of the Weibull shape parameter on the <math>pdf</math>.]]





Revision as of 21:46, 9 February 2012

The Effect of β on the [math]\displaystyle{ pdf }[/math]

The following Figure shows the effect of different values of the shape parameter, β, on the shape of the [math]\displaystyle{ pdf }[/math]. One can see that the shape of the can take on a variety of forms based on the value of β.

The effect of the Weibull shape parameter on the [math]\displaystyle{ pdf }[/math].


For [math]\displaystyle{ 0\lt \beta \leq 1 }[/math]:

  • As t→0 (or γ), f(t)→∞.
  • As T→∞, f(t)→0.
  • f(t) decreases monotonically and is convex as increases beyond the value of γ.
  • The mode is non-existent.

For β > 1:

  • f(t) = 0 at (or γ).
  • f(t) increases as [math]\displaystyle{ t\rightarrow \tilde{T} }[/math] (the mode) and decreases thereafter.
  • For β < 2.6 the Weibull [math]\displaystyle{ pdf }[/math] is positively skewed (has a right tail), for 2.6 < β < 3.7 its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal [math]\displaystyle{ pdf }[/math] , and for β > 3.7 it is negatively skewed (left tail). The way the value of β 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 β = 0.999, f(0) = ∞, but for β = 1.001, f(0) = 0. This abrupt shift is what complicates MLE estimation when β is close to one.