< 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, FX(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.