The Generalized Waring Distribution. Part I

The Generalized Waring Distribution is the hypergeometric distribution whose generating function is given by CF(a, k, p + a+ k, A), a 0, k > 0, p>O, C= pEk/(p +a)[kJ. For certain values of the parameters a, k it has extremely long tails; indeed all the moments can be infinite. (This need not be the case; where the first four moments are finite, it has been found useful in dealing with accident distributions.) In Part I various cases are distinguished, corresponding to special values of the three parameters. General formulae for the factorial moments, also /1, /2, are given as well as the forms these take in special cases. The continuous analogue of the discrete distribution is defined. In general, it is Pearson's Type VI though Types III, IV and V can occur in particular cases. A table of P/, P. is given and discussed for all combinations of the values p=8, 16, 24, 32, oo, qa = al(a + p), qk= kl(k + p) = 1, , ,i, 1. The mode of the distribution is obtained and its properties discussed.