ABSTRACT
Corisiclcr the particrllar case of I L distinguislra.k~l~ urns (ul -- ?l, , 7 1 , = 0, i -- 2 , 3, ..., n) and let be thc ~nnnber of h l l s distributed in the ;jth urn; j - 1, 2, ..., r ~ . 'I'hc randosn v;~.~.ii~bles X,, j - 1, 2, ..., r , arc called rantloin occupancy rlurrlbers. The reduction or' the joint distribution of the raiidoni occuparlcy rr~irnhers Z1 , X 2 , ... , X,, to a joint. coiitlitional distribution of intlcpcndcrit rarldorn variables X1 , Xz , ... , X,, giverr that S,, - X 1 $- X2 + . . . f - 7 1 1 is it po~verful technique in the tlcrivatio~i ancl st~itly of the distril~ution of the nrmilw K, ol11rns occupiul by i balls ea.c:h. I3arton arid David (1959), cor~sitlcrir~g a supply of balls ra.r~tlomly distrihutctl in n distinguishable 111.11s and assuming that the riurrrhcr X of ba.lls distributed i l l itriy specific llrn (X, - X , j -- 1, 2, ..., 71) is a random v;irial)le obeying ;t Poisson. binomial or negalive binorr~ial aw, clerivetl the p~.ob;hility funlct,ion and tht: fxtoriitl rnorr~c~~ts of the rlurr~her KO of cnipty urrls: in tllcsc three cases. givtn that S,, L ' I ~ I . ; these c1istril)utions arc thc classical, the 1.estrjct~t1 itrid the pscutio-contagiolls occupancy ctistrii.)lstions, respectively. It is m i t l i noticing that the ilss~m~pt,ion that X obeys a Po i s so~~ law is equivalent t,o the assllrription that t,lie ~ r r balls dis l r ihted in the u rw are dist,irigr~isllahlc. Ch;tritl;trirl)idcs (1986, 1997) derived, in the general case P(X = X) -- q,? 1. = 0,1 ,2 ; .... the probability fln~ction ancl factorial rilor~lerlts of the riurribe~. K = 71 KO of occupied urns and more gerlcrally of the i nm~bc~ . lii of urns occlqiccl by i 1)illls CRCEI, given that S;, -- m. llolst (1980), cor~sideriiig a si~pply of 'r tliff'crcr~t kiritls of I-mlls randomly clistribut,ctl in ~r tlistinguishal.)le urns antl assinning that the riunrlter X, , , of balls O S thc ,ith kind ilistribilted in the ,jt,h urn is ii random vari;tltlc ohcyirig ;l zcro-orie Bcrr~oulli law intleper~tler~t of i , tlcrived a representation of the characteristic fimctior~ of the rmnhcr K[, of empty u~ . i~ s , given that E; - y, balls of the 'ilh kind are distributed in the 17, urns, i -- 1, 2, ..., 1.. This representatiou is ~ised to stucly the asymptotic distril)utior~ of KO when 11, --) m. J4lrtller, the exact probability furrction and the fa.cloria1 rnornents of I<[, given that ): - l i = 1, 2 ; ... , r , were dcduccd. The assumption that X,,, ohcys a zero-one Bernoulli law is cquiv;tlcnt to the assurr~ptior~ that thc capacity of each urn is limited to one ball fro111 each kintl.
