Template:Characteristics of the exponential distribution alta

Characteristics
The characteristics of the 1-parameter exponential distribution can be exemplified by examining its parameter, lambda, $$\lambda ,$$  and the effect lambda has on the  $$pdf$$, reliability and failure rate functions.

Effects of $$\lambda $$ on the pdf

 * •	The scale parameter is $$\tfrac{1}{\lambda }$$.
 * •	As $$\lambda $$  is decreased in value, the distribution is stretched out to the right, and as  $$\lambda $$  is increased, the distribution is pushed toward the origin.
 * •	This distribution has no shape parameter as it has only one shape, i.e. the exponential. The only parameter it has is the failure rate, $$\lambda $$.
 * •	The distribution starts at $$T=0$$  at the level of  $$f(T=0)=\lambda $$  and decreases thereafter exponentially and monotonically as  $$T$$  increases, and is convex.
 * •	As $$T\to \infty $$,  $$f(T)\to 0$$.
 * •	This $$pdf$$  can be thought of as a special case of the Weibull  $$pdf$$  with  $$\beta =1$$.



Effects of $$\lambda $$ on the Reliability Function

 * •	The 1-parameter exponential reliability function starts at the value of 1 at $$T=0$$ . It decreases thereafter monotonically and is convex.
 * •	As $$T\to \infty $$,  $$R(T\to \infty )\to 0$$.

Effects of $$\lambda $$  on the Failure Rate Function

The failure rate function for the exponential distribution is constant and it is equal to the parameter $$\lambda $$.