Template:Characteristics of the gamma distribution: Difference between revisions

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===Characteristics of the Gamma Distribution===
#REDIRECT [[The_Gamma_Distribution]]
 
Some of the specific characteristics of the gamma distribution are the following:
 
For  <math>k>1</math> :
:• As  <math>t\to 0,\infty </math>  ,  <math>f(t)\to 0.</math>
:• <math>f(t)</math>  increases from 0 to the mode value and decreases thereafter.
:• If  <math>k\le 2</math>  then  <math>pdf</math>  has one inflection point at  <math>t={{e}^{\mu }}\sqrt{k-1}(</math>  <math>\sqrt{k-1}+1).</math>
:• If  <math>k>2</math>  then  <math>pdf</math>  has two inflection points for  <math>t={{e}^{\mu }}\sqrt{k-1}(</math>  <math>\sqrt{k-1}\pm 1).</math>
:• For a fixed  <math>k</math> , as  <math>\mu </math>  increases, the  <math>pdf</math> starts to look more like a straight angle.
:• As  <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.</math>
 
[[Image:BSpdf1.png|center|250px| ]]
 
For  <math>k=1</math> :
:• Gamma becomes the exponential distribution.
:• As  <math>t\to 0</math>  ,  <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
:• As  <math>t\to \infty ,f(t)\to 0.</math>
:• The  <math>pdf</math>  decreases monotonically and is convex.
:• <math>\lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}</math>  .  <math>\lambda (t)</math>  is constant.
:• The mode does not exist.
 
[[Image:BSpdf2.png|center|250px| ]]
 
For  <math>0<k<1</math> :
:• As  <math>t\to 0</math>  ,  <math>f(t)\to \infty .</math>
:• As  <math>t\to \infty ,f(t)\to 0.</math>
:• As  <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.</math>
:• The  <math>pdf</math>  decreases monotonically and is convex.
:• As  <math>\mu </math>  increases, the  <math>pdf</math>  gets stretched out to the right and its height decreases, while maintaining its shape.
:• As  <math>\mu </math>  decreases, the  <math>pdf</math>  shifts towards the left and its height increases.
:• The mode does not exist.
 
[[Image:BSpdf3.png|center|250px| ]]

Latest revision as of 07:43, 8 August 2012

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