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

(5) |

For exponentially distributed with scale parameter :

(6) |

For gamma distributed with shape parameter and scale parameter :

In , we used the asymptotic behavior of the gamma distribution

For normal distributed with mean and variance :

(8) |

erfc

that holds for and (we used ).
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 .

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