A review on recent generalizations of exponential distribution

Abstract

The Exponential distribution is the probability distribution of the time between two consecutive events in a Poisson point process, that is, it is a process in which the events occur continuously and independently at a constant average rate. The distribution is a limit of the scaled Beta distribution, and it is the only continuous probability distribution that has a constant failure rate. It has the maximum entropy probability distribution for a random variate X which is greater than or equal to zero, for which the expectation of X is fixed. The distribution is a particular case of the Gamma distribution, and it can also be seen as the continuous analogue of the Geometric distribution, it also possesses the lack of memory property, which states that the “process does not remember what has happened until now and the distribution of the waiting time, given that it has already exceeded some amount of time, has the same Exponential distribution form”. The distribution was referred to by Kondo1 as Pearson’s type X distribution when the sampling of standard deviation was discussed.