Let us study the tail distributions of three related distributions. The asymptotic decay rate measures the thickness of a random variable's tail distribution. It is defined by
(1) |
For exponentially distributed with scale parameter
:
(2) |
For gamma distributed with shape parameter
and scale parameter
:
In
For normal distributed with mean
and variance
:
(4) |
The asymptotic decay rates for the exponential distribution, gamma distribution, and normal distribution are given in ,
, and
, respectively. We see that the decay rate
of the normal distribution is always maximal, i.e.
, whereas the decay rates of the exponential distribution and the gamma distribution depend on the scale parameter
.
By normalizing the mean (
) by setting
and
, we can compare the decay rates of the exponential distribution and gamma distribution: then, the decay rate of the exponential distribution is
and of the gamma distribution is
. That is, keeping the mean normalized, the gamma distribution decays slower than the exponential distribution with shape parameters
, and faster than the exponential distribution with
. With
, the distributions coincide. When
, the gamma distribution approaches the normal distribution and
.
References: