On Characterizion of Distributions

Tamhankar(l967) studied a probability distribution on) an n-dirnensionalEuclidean space can be specified by giving either the joint distribution of the usual’ rectangular coordinates or by the joint distribution of the polarcoordinates. The object of this jo 2btain a aracterjzation oforn ality .in terms of independence of these coordinates under suitable regularityconditions. Further, if we slightly relax these conditions we get a new class of distributions which includes the normal distribution. ~I:!.eela 1970) studied a ch”a~cteriz~~o!!.”,2f the. doyble Poisson distri~.uti9n isobtained. The result can be directly extended to obtain a characterization of the multiple Poisson distribution in more than two dimensions. Shanbhag (1970) gave an interesting characterization of the exponcntiai andthe geometric distribution in terms of conditional expectations. In this thesis the same method is further used to obtain a characterization for somedistributions related to operations research and reliability.Ahsanullah and Rahman(1972) presented A necessary and sufficient conditionbased on order statistics that a positive random variable having an absolutely’ continuous probability distribution (with respect to Lebesgue measure) will bc . exponential given