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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:
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| For <math>k>1</math> :
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| :• As <math>T\to 0,\infty </math> , <math>f(T)\to 0.</math>
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| :• <math>f(T)</math> increases from 0 to the mode value and decreases thereafter.
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| :• 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>
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| :• 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>
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| :• For a fixed <math>k</math> , as <math>\mu </math> increases, the <math>pdf</math> starts to look more like a straight angle.
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| As <math>T\to \infty ,\lambda (T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
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| [[Image:ldaGD10.1.gif|thumb|center|400px| ]] | |
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| For <math>k=1</math> :
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| :• Gamma becomes the exponential distribution.
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| :• As <math>T\to 0</math> , <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
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| :• As <math>T\to \infty ,f(T)\to 0.</math>
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| :• The <math>pdf</math> decreases monotonically and is convex.
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| :• <math>\lambda (T)\equiv \tfrac{1}{{{e}^{\mu }}}</math> . <math>\lambda (T)</math> is constant.
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| :• The mode does not exist.
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| [[Image:ldaGD10.2.gif|thumb|center|400px| ]]
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| For <math>0<k<1</math> :
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| :• As <math>T\to 0</math> , <math>f(T)\to \infty .</math>
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| :• As <math>T\to \infty ,f(T)\to 0.</math>
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| :• As <math>T\to \infty ,\lambda (T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
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| :• The <math>pdf</math> decreases monotonically and is convex.
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| :• As <math>\mu </math> increases, the <math>pdf</math> gets stretched out to the right and its height decreases, while maintaining its shape.
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| :• As <math>\mu </math> decreases, the <math>pdf</math> shifts towards the left and its height increases.
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| :• The mode does not exist.
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| [[Image:ldaGD10.3.gif|thumb|center|400px| ]]
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