ISSN:
1572-9052
Keywords:
Characterization
;
exponential distribution
;
gamma distribution
;
geometric distribution
;
Poisson process
;
renewal process
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Given two independent positive random variables, under some minor conditions, it is known that fromE(Xr∥X+Y)=a(X+Y)r andE(Xs∥X+Y)=b(X+Y)s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t≥0} be a renewal process, with {S k, k≥1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, k≤n, ifE(S k r ∥A(t)=n)=atr andE(S k s ∥A(t)=n)=bts, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t≥0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01720591
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