Template:Characteristics of the gamma distribution

Characteristics of the Gamma Distribution

Some of the specific characteristics of the gamma distribution are the following:

For [math]\displaystyle{ k\gt 1 }[/math] :

• As [math]\displaystyle{ t\to 0,\infty }[/math] , [math]\displaystyle{ f(t)\to 0. }[/math]

• [math]\displaystyle{ f(t) }[/math] increases from 0 to the mode value and decreases thereafter.

• If [math]\displaystyle{ k\le 2 }[/math] then [math]\displaystyle{ pdf }[/math] has one inflection point at [math]\displaystyle{ t={{e}^{\mu }}\sqrt{k-1}( }[/math] [math]\displaystyle{ \sqrt{k-1}+1). }[/math]

• If [math]\displaystyle{ k\gt 2 }[/math] then [math]\displaystyle{ pdf }[/math] has two inflection points for [math]\displaystyle{ T={{e}^{\mu }}\sqrt{k-1}( }[/math] [math]\displaystyle{ \sqrt{k-1}\pm 1). }[/math]

• For a fixed [math]\displaystyle{ k }[/math] , as [math]\displaystyle{ \mu }[/math] increases, the [math]\displaystyle{ pdf }[/math] starts to look more like a straight angle.

As [math]\displaystyle{ t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}. }[/math]

For [math]\displaystyle{ k=1 }[/math] :

• Gamma becomes the exponential distribution.

• As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ f(T)\to \tfrac{1}{{{e}^{\mu }}}. }[/math]

• As [math]\displaystyle{ T\to \infty ,f(T)\to 0. }[/math]

• The [math]\displaystyle{ pdf }[/math] decreases monotonically and is convex.

• [math]\displaystyle{ \lambda (T)\equiv \tfrac{1}{{{e}^{\mu }}} }[/math] . [math]\displaystyle{ \lambda (T) }[/math] is constant.

• The mode does not exist.

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

• As [math]\displaystyle{ t\to 0 }[/math] , [math]\displaystyle{ f(t)\to \infty . }[/math]

• As [math]\displaystyle{ t\to \infty ,f(t)\to 0. }[/math]

• As [math]\displaystyle{ t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}. }[/math]

• The [math]\displaystyle{ pdf }[/math] decreases monotonically and is convex.

• As [math]\displaystyle{ \mu }[/math] increases, the [math]\displaystyle{ pdf }[/math] gets stretched out to the right and its height decreases, while maintaining its shape.

• As [math]\displaystyle{ \mu }[/math] decreases, the [math]\displaystyle{ pdf }[/math] shifts towards the left and its height increases.

• The mode does not exist.