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
For exponentially distributed with scale parameter :
For gamma distributed with shape parameter and scale parameter :
, we used the asymptotic behavior of the gamma distribution
For normal distributed with mean and variance :
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 .