< Probability distributions list < *Exponential-type distribution *

The** exponential-type distribution** is a broad class of probability distributions that includes many common probability distributions such as the exponential distribution, the gamma distribution, the Gumbel distribution, the log-normal distribution, the normal distribution and the Weibull distribution.

Although sometimes used interchangeably, the exponential distribution and exponential-type distribution are not the same thing; the exponential distribution is a sub-type characterized by a constant hazard rate. Although the exponential-type distribution can be used to model a wide variety of phenomena, the exponential distribution is simpler and often easier to work with.

The exponential-type distribution can model a wide variety of phenomena. It is often used in reliability engineering, where it can be used to model a system’s time to failure. In finance, it can be used to model the distribution of asset prices.

## Exponential-type distribution properties

Exponential-type probability density functions (PDF) have the form [1]:

Where A(·), B(·), C(·), and D(·) are arbitrary functions.

The cumulative distribution function (CDF) is [2]

where *g*(*x*) is an increasing function of* x*. In other words, *F _{X}*(

*x*) approaches 1 at least as fast as an exponential distribution.

We can also define the exponential-type distribution in terms of derivatives [3]. Suppose that *X* is a random variable with CDF *F*, with an infinite upper end point and such that *F*^{(j)}(*x*), *j* = 1, 2, …, exists. Then a distribution function *F* is of the exponential type, if for large *x*,

This definition states that a distribution function *F* is of the exponential type if the ratio of the successive derivatives of F(x) converges to a constant as *x* approaches infinity. This means that the exponential-type family shares a common asymptotic behavior.

Exponential-type distributions have finite moments of all orders [3].

## Exponential-type functions

The exponential-type distribution and the exponential-type function are closely related concepts in probability theory.

While the exponential-type distribution is a family of probability distributions that share a common asymptotic behavior, the** exponential-type function** represents the PDF of the exponential-type distribution. It can be used to calculate the probability of a random variable from the exponential-type distribution taking on a specific value.

## References

[1] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[2] Sobczyk, K. & Spencer, B. Random Fatigue: From Data to theory. (1992) Academic Press.

[3] Csorgo, M. and Krishnaiah, P. (2010). From Finite Sample to Asymptotic Methods. Cambridge University Press.