Template:Loglogistic distribution characteristics: Difference between revisions

Latest revision as of 09:58, 9 August 2012

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-==Distribution Characteristics==
+#REDIRECT [[The_Loglogistic_Distribution]]
-For  <math>\sigma >1</math> :
-:*	 <math>f(t)</math>  decreases monotonically and is convex. Mode and mean do not exist.
-For  <math>\sigma =1</math> :
-:*	 <math>f(t)</math>  decreases monotonically and is convex. Mode and mean do not exist. As  <math>t\to 0</math> ,  <math>f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>
-:*	As  <math>t\to 0</math>  ,  <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>
-For  <math>0<\sigma <1</math> :
-:*	The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
-:*	The  <math>pdf</math>  starts at zero, increases to its mode, and decreases thereafter.
-:*	As  <math>\mu </math>  increases, while  <math>\sigma </math>  is kept the same, the  <math>pdf</math>  gets stretched out to the right and its height decreases, while maintaining its shape.
-:*	As  <math>\mu </math>  decreases,while  <math>\sigma </math>  is kept the same, the  ..  gets pushed in towards the left and its height increases.
-:*	 <math>\lambda (t)</math>  increases till  <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}</math>   and decreases thereafter.  <math>\lambda (t)</math>  is concave at first, then becomes convex.
-[[Image:WB.15 loglogistic pdf.png|center|250px| ]]