RANDOM DISTRIBUTIONS WITH AN APPLICATION TO TELEPHONE ENGINEERING
Robert M. Fortet · 1956
General Poisson distributionsRecently, A. Blanc-Lapierre and I remarked that we were not aware of any definition of a general Poisson distribution, that is to say, of a Poisson distribution not, as usual, on the straight line or on some Euclidean space, but on a perfectly general space.Such a definition may be useful, and can be given in the following obvious way.Let X be any space of elements x, B a Borel field of subsets e of ;, and m(e) a meas- ure on B (not necessarily bounded or finite).A random family F of elements of J is a Poisson distribution on T [with respect to eB and m(e)] if, M(e) being the number of elements of F belonging to e E A, we have the following properties [1]:1) If m(e) 0, the distribution on e of the k elements of F belonging to e is statistically equivalent to the choice at random, independently, of k elements x on e, with Pr{x E e'} = m(e')/m(e), where e' is any subset of e belonging to A.2) Let et be a family of sets belonging to B, 0 _ t T, at least one element of F belongs to ei.