Random Alms

Abstract

1. Statement of the problem. Consider the problem of distributing one pound of gold dust at random among a countably infinite set of beggars. Let the beggars be enumerated and let the procedure for distribution be as follows: the first beggar is given a random portion of the gold; the second beggar gets a random portion of the remainder;-* and so on ad infinitum. In this description the phrase "random portion" occurs an infinite number of times: it seems reasonable to require that it have the same interpretation each time. To be precise: let xi (j 0, 1, 2, * *) be the amount received by the jth beggar. Let the distribution of xo be given by a density function p(X): (1) p(X) O, O < X < 1; (2) fP(X)dX = 1; b (3) P(a < xo < b) = p(X) dX, 0 < a < b ? 1. After the first beggar has received his alms and the amount of gold dust left is ,u (i.e. xo = 1-j,), the value of xi will be between 0 and A. The uniformity requirement mentioned above means that the proportion of Au that the second beggar is to receive is again determined by the probability density p: in other words the conditional probability that xi be between XAu and (X + dX),u, given that xo = 1-Iu, is p(X) dX. In symbols: bi ~~~~~~~b (4) P(aA < xi< bpIxo-d,u)=fp()dX. Writing a = aju, ,B = bAu, (4) becomes (5) P(ae < XI < x < b I xi = 1-)=f P) dX, a < b < ?A. This assumption completely determines (in terms of p) the joint distribution of the whole infinite sequence { xO, xl, X2,