Exponential and Hypoexponential Distributions: Some Characterizations

George P Yanev

doi:10.3390/math8122207

Abstract

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

Highlights

Introduction and Main ResultsSums of exponentially distributed random variables play a central role in many stochastic models of real-world phenomena
Hypoexponential distribution is the convolution of k exponential distributions each with their own rate λi, the rate of the ith exponential distribution
If we have a k + 1 state process, where the first k states are transient and the state k + 1 is an absorbing state, the time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in state 1 and move skip-free from state i to i + 1 with rate λi until state k transitions with rate λk to the absorbing state k + 1

Summary

Introduction and Main Results

Sums of exponentially distributed random variables play a central role in many stochastic models of real-world phenomena. N are not all identical, is called (general) hypoexponential distribution (see [1,2]) It is absolutely continuous and we denote by gn its density. If Zi are independent and identically distributed random variables and their sum has Erlang distribution, the common distribution is exponential. Xn , for fixed n ≥ 2, are independent and identically distributed as a random variable X with density f , f ( x ) = λe−λx , x > 0. In case of unit exponential distribution, (2) can be written as follows: Sn := μ1 X1 + μ2 X2 + · · · + μn Xnj has density gn ( x ) := ∑ F μ j =1 j x μj (6).

Auxiliaries

Concluding Remarks