Template:Characteristics of the exponential distribution alta

Characteristics

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

Effects of λ on the pdf

• The scale parameter is [math]\displaystyle{ \tfrac{1}{\lambda } }[/math] .
• As λ is decreased in value, the distribution is stretched out to the right, and as λ 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). Its only parameter is the failure rate, λ .
• The distribution starts at T = 0 at the level of f(T = 0) = λ. It decreases thereafter exponentially and monotonically as T increases, and it is convex.
• As [math]\displaystyle{ T\to \infty }[/math] , [math]\displaystyle{ f(T)\to 0 }[/math].
• This pdf can be thought of as a special case of the Weibull pdf with β = 1.

Effects of λ 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 [math]\displaystyle{ T\to \infty }[/math] , [math]\displaystyle{ R(T\to \infty )\to 0 }[/math].

Effects of λ on the Failure Rate Function

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