Effect of beta on the Weibull pdf plot - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2015-06-25T19:48:47Z

Redirected page to The Weibull Distribution

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	[[~~Category: For Deletion~~]]		#REDIRECT [[The Weibull Distribution]]

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2012-07-24T06:22:15Z

Replaced content with 'Category: For Deletion'

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	~~\| valign="middle" align="left" bgcolor=EEEDF7\|~~[[~~Image~~: ~~Weibull-Examples-banner.png\|400px\|center]]~~
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	~~<br>~~
	~~[[Example: The Effect of Beta on the Weibull pdf]]~~
	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 > 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 < 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.

Lisa Hacker at 11:49, 17 July 2012

2012-07-17T11:49:17Z

← Older revision		Revision as of 11:49, 17 July 2012
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	<br>		<br>
	[[Example: The Effect of Beta on the Weibull pdf]]		[[Example: The Effect of Beta on the Weibull pdf]]
	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>.		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>.]]		[[Image:WB.8 weibull pdf.png\|center\|400px\| The effect of the Weibull shape parameter on the <math>pdf</math>.]]

Nicolette Young: Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image: Weibull-…'

2012-04-02T21:50:26Z

Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image: Weibull-…'

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{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"
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| valign="middle" align="left" bgcolor=EEEDF7|[[Image: Weibull-Examples-banner.png|400px|center]]
|}
 
[[Example: The Effect of Beta on the Weibull pdf]]
The following Figure shows the effect of different values of the shape parameter, β, 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 β.

[[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 ''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> t\rightarrow \tilde{T} </math> (the mode) and decreases thereafter.
:*For β < 2.6 the Weibull <math>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>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.