Characteristics of the 1-parameter Exponential Distribution

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 .
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. The only parameter it has is the failure rate, λ.
The distribution starts at T = 0 at the level of f(T = 0) = λ and decreases thereafter exponentially and monotonically as T increases and is convex.
As T , f(T) 0.
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 T , R(T ) 0.

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.