Template:Characteristics of the gamma distribution

Characteristics of the Gamma Distribution
Some of the specific characteristics of the gamma distribution are the following:

For $$k>1$$ :
 * •	As $$t\to 0,\infty $$ ,  $$f(t)\to 0.$$
 * •	 $$f(t)$$ increases from 0 to the mode value and decreases thereafter.
 * •	If  $$k\le 2$$  then  $$pdf$$  has one inflection point at  $$t={{e}^{\mu }}\sqrt{k-1}($$   $$\sqrt{k-1}+1).$$
 * •	If  $$k>2$$  then  $$pdf$$  has two inflection points for  $$t={{e}^{\mu }}\sqrt{k-1}($$   $$\sqrt{k-1}\pm 1).$$
 * •	For a fixed $$k$$, as  $$\mu $$  increases, the  $$pdf$$ starts to look more like a straight angle.
 * •	As $$t\to \infty ,\lambda (t)\to \tfrac{1}.$$



For $$k=1$$ :
 * •	Gamma becomes the exponential distribution.
 * •	As $$t\to 0$$ ,  $$f(T)\to \tfrac{1}.$$
 * •	As $$t\to \infty ,f(t)\to 0.$$
 * •	The $$pdf$$  decreases monotonically and is convex.
 * •	 $$\lambda (t)\equiv \tfrac{1}$$ .  $$\lambda (t)$$  is constant.
 * •	The mode does not exist.



For $$0<k<1$$ :
 * •	As $$t\to 0$$ ,  $$f(t)\to \infty .$$
 * •	As $$t\to \infty ,f(t)\to 0.$$
 * •	As $$t\to \infty ,\lambda (t)\to \tfrac{1}.$$
 * •	The $$pdf$$  decreases monotonically and is convex.
 * •	As $$\mu $$  increases, the  $$pdf$$  gets stretched out to the right and its height decreases, while maintaining its shape.
 * •	As $$\mu $$  decreases, the  $$pdf$$  shifts towards the left and its height increases.
 * •	The mode does not exist.