## December – Asymptotic decay rate of a probability distribution

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)

where denotes the probability of an event. Let us calculate the decay rates for the exponential distribution, gamma distribution, and normal distribution.

For exponentially distributed with scale parameter :

 (6)

For gamma distributed with shape parameter and scale parameter :

 (7)

In , we used the asymptotic behavior of the gamma distribution

For normal distributed with mean and variance :

 (8)

where in , we used the inequality

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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