Let be ( partly ) independent , exponentially distributed random variables with rates and let be probabilities whose sum equals 1. Then the random variable is called hyperexponentially distributed if it has the following probability density :
Classification and remarks
In the case of an exponential distribution, the coefficient of variation (standard deviation divided by the expected value) is equal to 1. The term “hyper” -exponential comes from the fact that the coefficient of variation here is greater than 1 (if different occur). In contrast to this, it is less than 1 for the hypoexponential distribution . While the exponential distribution is the continuous analogue to the geometric distribution , the hyperexponential distribution is not an analogue to the hypergeometric distribution . The hyperexponential distribution is an example of a mixed distribution .
The utilization of an Internet connection via which either (with probability and rate ) Internet telephony or (with probability and rate ) file transfers run can serve as an application example , whereby . The total load is then distributed hyperexponentially.
A given probability distribution, including end-load distributions , can be approximated by a hyperexponential distribution by recursively fitting different time scales ( ) using the so-called Prony method.
properties
The linearity of the integral results in:
and
With the help of the displacement theorem, the variance results:
Unless they are all the same size, the standard deviation is greater than the expected value.
↑ LN Singh, GR Dattatreya: Estimation of the Hyperexponential Density with Applications in Sensor Networks . In: International Journal of Distributed Sensor Networks . 3, No. 3, 2007, p. 311. doi : 10.1080 / 15501320701259925 .