Department of Mathematics Temple University~ Philadelphia

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1 lsupported by the U.S. Army Research Office, Durham, under Contract No. DAHC0471C Supported by the Air Force Office of Scientific Research under Grant AFOSR EXTENDED OCCUPANCY PROBABILITY DISTRIBUTION CRITICAL POINTS by N.L. Johnson l Department of Statistics University of North Carolina at Chapel Hill S. Kotz 2 and R. Srinivasan Department of Mathematics Temple University Philadelphia Institute of Statistics Mimeo Series No. 934 June, 1974

2 EXTENDED OCCUPANCY PROBABILITY DISTRIBUTION CRITICAL POliHS By.J.L. Johnson, University of Horth Carol ina at Chapel Hill, S. Kotz and R. Srinivasan, Temple University, Philadelphia 1. Introduction In many instances of application of statistics, the underlying probabilistic problem can be described in terms of the following urn model: N balls are randomly allocated to K cells (so that each ball has probability 11K of falling into the ith cell, i = 1, 2,.., K). Each ball has probability p of staying in its cell and 1 P of I1falling through l1 (and hence not being available to 'fill' a cell). The random variable of interest here is the number x of 'occupied' cells, or equivalently the number K x of 'empty' cells. Several examples of situations leading to this urn model in biology, nuclear phyics and other fields are given in Feller (1968) and Harkness (1969). Our purpose here is to tabulate critical points of the distribution for various values of N, K and p. The case when p = 1 has already been treated at length by Nicholson (1961), and we shall therefore not include it here. A description of our tables is given in the next section. They are set out in a similar form to Nicholson's tahle. Critical roints corresponding to some of the values of Nand Knot covered in our tables can be approximated using certain limit theorems. This is discussed in Section 3. The classical occupancy distribution (corresponding to p = 1) gives the probability of exactly x occupied cells as

3 2 (1) Hl(x;K,N) = C\..: / x L (l)(xi) i=o J... tr N = (x = 0,1,., K) For general p, the number of balls not falling through is distributed binomially with parameters N,p and so the extended occupancy distribution is the corresponding mixture of classical occupancy distributions. The probability, tlp(x;k,n), of exactly x occupied cells is then the expected value of HI (x;k,j) with J distributed binomially with parameters H,p. Hence (2) H (x; K,N) P = These probabilities satisfy the recursion relation (3) KHp(x,K,N) = P(K+Ix)hp(xI;K,Nl) The rth factorial moment is + {px + (lp)k} H (x;k,ni) P (4) ll(r) = = = = where t(r) = t(tl)... (tr+l).

4 3 From (4) we obtain pressions for the expected value: (5.1) N K{l (lp/k) } and the variance (5.2). N N N K(Kl) {l2 (lp/k) 0+ (12p/K) } + K{l (lp/k) } = i( {lo:. (l_p/k)n} (l_p/k)n _ K(Kl) {(lp/k) 2N _ (1_2p/K)N} The rth factorial moment of the number of empty cells has the simpler form (6) K(r)(l _ pr/k)n (Harkness (1969)). 2. Description of Our Tables Using (3), we can find, for given K, X, P and lower tail area a, X the smallest N, say N I, such that Pr[x X K, Nl,p] = L H (x,k,n) a. These are presented in Table I for K = 2 (1) 20, P = 0.75,0.80,0.90, 0.95, 0.99, and a = 0.01, 0.025, 0.05, 0.10, and N l up to 60, along with the the actual tail probabilities a l = P [x X I K,N 1,p]. (For example, the entry N I = 17, a l = 0.019, corresponding to p =.99, K = 8, X = 5 and a = in Table I means that Pr[x 5 I 8, 17, 0.99] = 0.019, Pr[x 5 I 8, N, 0.99] > for N < 17, and that Pr[x 5 I 8, N, 0.99] < for N > 17). There is one exception to this system. If there is a value N 1 such that Pr[X x I K, N I, p] lies between a and a this is the value given in the tables. The corresponding value of a l is given as a +. x=o p

5 4 Similarly, using the recursion relations; for the same values of K, X, P and Ct we have tabulated the upper tail critical values N 2 and the corresponding probabilities Ct 2 in Table II. Here, for given K, X, P and Ct, N Z is the largest value of N such that Pr[x X K, N, il] Ct, and Ct = Pr[x X I K, N, 2 p]. (For example, 2 =.021, corresponding to p = 0.80, K = 10, X = 9 the entry N 2 = 12, Ct 2 in Table II means that p[x 9 I 10, 12, 0.80] = 0.021, 10, N, 0.80] < for N $' 12, and that Pr[x 9Il0,N.0.80] ;> for N > 12,.) Dash marks () in table II indicate the nonexistence of N 2 for given K, X and Ct. Here again, as an exception if there is an N such that Ct < 2 P [x X I K, N, p] < Ct + r this + is the value given in the tables, with Ct given as Ct 2 In both lower and upper tail tables the entry > 60 is used when the exact value is beyond the range of calculations we have carried out. 3. Approximations Harkness (1969) has suggested that a binomial with parameters K, 1 (l_p/k)n would give a good approximation to the distribution (2). This approximation gives the correct mean (see (5.1)) and is exact for K = 1. It would be expected to give good results for p small, since the mixing distribution of J (see Section 2) may then be approximated by a Poisson (with parameter Np), and as Harkness shows a mixture of classical occupancy distributions with such a Poisson mixing distribution is, in fact a Binomial distribution.

6 5 If the binomial approximation is good, then Molenaar's (1970) normal approximation to the binomial may be used to get approximate expressions for N l and N 2 Writing (7) wen) = 1 (l_p/k)n we have from Molenaar's 'quick work' approximation (1970, p. 110) for a.. ::; 0.05 J (8.1) (3.2) where 1 foo 2 (I;f.;) V exp( ) du = a.. a. T.e equations for a.'. > 0.05 are of similar form. J In order to solve equations of form AIlw(N) B/w(N) = V we write wen). 2 e = SIn = sin rp and get where (9) wen) T (when A 2 +B 2 < V 2 this equation is not applicable, but since A 2 +B 2 is about 2 4K while hhe largest V we need is 5.4, this imposes little restriction.)

7 6 Once from (7). w(n.) J has been found we can calculate the corresponding N. J Some trial calculations with K = 20 indicated that when p 0.75 this approximation gives values of N which are too big by some 24 l units and of N which are too small by some 12 units, at any rate for 2 a = Further asymptotic formulae will be found in SamuelCahn, (1974).

8 7 REFERENCES 1. F.N. David, (1950). Two combinatorial tests of whether a sample has come from a given population, Biometrika, 37, W. Feller, (1968). An Introduction to Probability Theory and its Applications s New York: John Wiley &Sons, Inc. 3. W.L. Harkness, (1969). The czassical occupancy problem revisited, Pennsylvania State University, Dept. Statistics, Technical Report No W. Molenaar, (1970). Approximations to the Poisson, Binomial and Hypergeometric Distribution Functions, Mathematical Centre Tracts No. 31, Math. Centrum, Amsterdam. 5. N.L. Nicholson, (1961). Occupancy distribution critical points, Biometrika, 48, Ester Samue1Cahn, (1974). Asymptotic distributions for occupancy and waiting time problems with positive probability of falling through the cells, Annals of Probability, 2,

9 .'_., co Lower tail critical points _...,_...._..' _._._._..._... p =.75 p =.75 '_._".' Lower tail probability level a. Lmrer tail probability tvei C _...,",,'".,.._..._._._.... _. N N N N N N a lit fj,j. N _._,..... _._ ( ' ' } , I "028 9.C ,..,..J.., Ol ' "01; _ h o 2 030i _

10 . 0'\ Lo tail critical E.oints _...'_ p =.75 p =.75._.,...,_... Lower tail probability level C't Lower tail proability level C't _., N 1 C't 1 N 1 C't 1 N 1 C't 1. ".. H 1 C't 1 _..._... N 1 <Xl 11 1 C't 1 iii C't 1 N"1 C't 1 _ ' , :> q l In a ' _

11 R....._._._..._ ".._... Lower tail critical J2..oints._.. p =.75 p =.75 Lower tail probability level Lower tail probability level 0 _._. K 11 N 1 1IJ N 1 N 1 N N 1 a 1.L 0.1 'I 1).1 1).1 '1 1).1 1).1 al..,_ "' 4_., = / q " "' r; l _ _ _ ') 14 9.) , Oll , , ,

12 ,.,..._ _. _ '" _._ Lower tail critical points..._ '.._..... _.._..._._._.. '"._.....,_.0_ P =.75 P =.75 Lower tail probability level a,_._,_._,,_._. LmTer tail probability level a _....; _..._,.. '..._._,...._ HI a N a N a 1 N 1 a 1 J 1 "'1 a N 1 1 a 1 HI a N a 1 1 1,_.._ _ '" _ >(.') > ) "' "' '.j " II '"' ".. u ,... _...

13 5 5. N Upper tail critical points. p =.75 p =.75 Upper tail probability level I), Upper tail probability level I), K X T( h X :JII 2 1),2 N 2 1),2 N a J'T ),2 1'1 a. ]\1 2 2 '2 1),2 N 2 1),2 N a ,.""., , 6 "..> 'T U '... _ <'" ,,,, 6 _. h ""' ' ,".. ' " 6 5 '" _ _ q , c , _..... c./ _.

14 t').l Yer tail critical point. p =.75 p =.75,' Upper tail probability level C't. Upper tail probability level C't. N _._ N 0. N lit C't. C't N 2 C't. N C't.. 2 "2 IJ ' _ N ' , '"' ) Pi o _. <"'" ", _ _ l _.. _. 0..' :' 7.. ' (3., C\,/ f) oll

15 ,..... 'd"..4 Yil critical points p =.75 p =.75 _.._,_...._..._,.. Upper tail probability level a Upper tail probability level a N 2 lx 2 N 2 lx 2 N 2 lx 2 l\t T,T 0. 2 N,., a 2 ]IT 0. 2 "2 ;;;: '2 N f5 17.e It) l "'" _. _..."" "' ".' r { _ ! f ' _ / , ,OIl lq , f J ,009 50,

16 _.._._._._. "_. _",,, _.,......_...'....._....' l/)...t.. p =.80._.... """'. 4 Lower tail probability level u _..,. _.._.. Lower tail ritica!. p?ints.. _. _ _ _.,... _..._ _... _._N....,_.._ _._ p =.80 _._.._...._._.. Lower tail probability level u... _..._...,.._._ _..,_.,... K X A 6 :( X N u 1 1 HI u N u N u 1 N._._.. _... 1 '\.. _.w_ _...,._.._ ,.. "1 N 1 u 1 TT u l N 1 '\ , J t:: "" _ , , " OOG u 9,

17 u '... A_._ 1; "' _._,...,..._....._...,._. 55 \0...i &ovrer_ ail_... _.... _... _.. critical points _._...._ _ _......_._,_... p =.80 l' =.80 Lower tail probability level a Lower tail probability level a N a N rt a !III a 1 :N a. IJ a 1 N 1 a 11 1 'I 1 a : let> l "' _ " ( , : l f"i ;; ,

18 ...,, r..1 J:,owe tail ri.!ical :e,.oints ,,....._,.,... _._._._._...,._..._. _... _... _......_._._.... _..... e.... p =.80 p = _,_._._ Lower tail probability level a Lower tail probability level..._,... "._... '''h._._....,..._.. _.._..._.._.._ _...._...._._.._._. _ _ W l 1 N l Ct 1 N ,,.._..._.... _... 1'1 '1 1\1 1 N l 1 II I l N ' _ q ll _ _ _ " ""., "0' JOO

19 ,./ 00...,_. Lme! t!lg c}'_ii point!.. _ _. L._,. _..._', _... _.,, "._......_,'._._.;,..._',. p =.80 p =.80,._..,._...._.._ Lower tail probability level (1 Lower tail probability level (1 _._._,,_._.._..._,.... _..._._._... _._._._,,'_. _..._. III lit (11 (11... N N N l 1 l (11 1'1 6\.T (11 (11 1 (11 1'1 "'1 '1..'>... _......_. _.._..,.._._, _.._.,.,_..._.,_..,_._ (11 N 1 (11 ".," ! , M ,, < >:;0,.."" >61 _..,.,.,.' "c'> ! ') , h r )! _ , "....., _ '"". ",,

20 )..l Upper tail critical points p =.80 p =.80 Upper tail probability level (l Upper tail probability level (l N 2 (l2 N 2 <X 2 JlT 2 (l2 N 2 (l2 T2 (l2 N \1"2 (l2 N 2 (l2 4 4 _. _ , CO" " ' I OlO Olf Oll '' '. 00' '" " '. " 'r """,...'. ".' "., (, , 7 "' '.. 'I.", <"' 8 "' 8.Oll , , _

21 ,.... o N Upper:. tail c_ritical poit:lts p =.80 p =.80 Upper tail probability level a Upper tail probability level a. N 2 0. N 2 2 (12 N 2 0. N 2 0. ',r 2 0. N N 2 L a" c:. ' u'2 1\T , Q " '., (1..../ "'"" / ,.. 6, , _ , () 9.010,_:> l ""' "' <'< 7,, O

22 N Upper tail critical POi1tS p =.80 p =.80._ Upper tail probability level a Upper tail probability level a._.,,..<_._., _. N a N a N' a a 1\T 2 a N 2 2 a }IT a 2 lj a " f l j _. < a./ : '?O

23 N N..._...._...._'_._.._....._.'.,. ow tai..j.:. critic:tl_ points._..._._... _ _._._,... _..._._..._.._.. _...._. p =.90 P =.90 Lower tail probability level a _._._ Lower tail probability level a. _._... '_. _._. M.... _.. _.,'W... _...._,.. '.....' N _'_._,... "_ N N _.. ' _..._... _.... _.. e. N 1 ell H 1 0. N 1 1 ell N '1 a H 1 1 ell,._ _.... """' '.O'._d. _._.., '" = ,' ':J O J , K;o Q "':..050

24 .._., 4 _,,_ _._ t<") N,.._..._.... Lower tail critical points _. _._ ,......'..._.._._..._.._._...._._._._...._. '.... p =.90 p =.90 Lower tail probability level a. L01rer tail probability level a _._._._'_._ "...'.,,..._.....,..,,_.....'.._.._'..._. N a N (l1 N 1 (l1 F (l1 1\T 'I '1 (l1 N a N (l1 N 1 (l1... _.._ :\ J ,. """" '.' 10 _ ) l 'r _ '

25 59 _. 4._..,.,. 57 '<:t N r...o_e!_j;_a...i_1_.!_t.i.caj..1'oint...._ p =.90 P =.90 Lower tail probability level a Lower tail probability level a....._,..._... CH _, '' "._c... _,_. _.,,,.. w_c_...!'i 1 (Xl 'tit a 1 N 1 a 1 N a 1 IJ a 1 N 1 a 1 H 1 (Xl N 1 (Xl ',", >O.....' ",0 18., "' l " II _..

26 ... 59,'.._._...._.... L!'l N ' _.#_._._...._....,... _._.,_..,.,.. LT_e! :t.i_1 c:rj::t.ia1_.:e9ints ' _. _.._.> _._ _.._ _ "' p =.90 p =.90 _...,. Lower tail probability level 0. Lower tail probability level 0. K X K X , _,,.._... '_.. _.._... '," _..._ HI Ct TiT 1 '1 0. N 1 1 Ct. N N 1 l 0. 1 HI 0. 1 '1 _._..,_..._.,,,...._...'.._.'..._...._. u._. #,_.._"'.< '..,...,. _..._ c lj (\ N 1 Ct , , ,/ _. _ I' rr

27 \0 N Upper tail crtica1 points p =.90 p ::;:.90 Upper tail probability level Upper tail probability level.._.... K X I( X :N 2 2 N 2 2 N 2 2 N r 2 N,., 2 JlT N t! c_ ' &". r ". 7.. o r',., _ ) l _ " Ij.."" _ OO' _.. r.' ,,..,..." , " '7 I """

28 _.,.,.' 9 l" N Upper tail critical point p =.90 p = 90 Upper tail probability level u Upper tail probability level a N u N u N u N Ct l'j 2 u N u lit u 2 N u '" ""' l1.t IS J " <.' _. '" a..03!t ).j , _../ <, _

29 00 N UFper tail critical points p =.90 p = 90 Upper tail probability level a Upper tail probability level Ct._ J N Ct N Ct N a N Ct J'2 Ct N 2 2 Ct N Ct N Ct G J , , ""

30 _. 0'1 N _..,,_...'..... LmTex: ail_ E..r.J:_ical poii_i?.,'_. _ _._....,..,.. _._...,.... _._....._..._..._ p =.95 p =.95,.... Lower tail probability level a Lower tail probability level..._..._._.._..._..._,..._... _. _.....,._...,.._..._..._... _. '*_.,_..., K X T' v A _,_. _.._. 1'..._..._._...' ,_ _. N WI (Xl "'1 N 1 1 (Xl HI (Xl "J N 1 1 (Xl N (Xl N (Xl.., _.. _...._'_._ 'c _ ll ' ';) , J _ ,.

31 ..._..._ u_._._.. _....,.._._. 0 _.._....,._._._. _..._._,.,,....' _,._. n. Lower tail critic'u E.oints,..._..._._.._,...._._...,. p =.95 P =.95 _.._.... _.._,' _..._... Lower tail probability level a,._......_..... _ _. Lower tail probability level a.._.._...._.._a _.._. _...,....,._ N a JIT a N a N a l 1 l 1'T a ';IT a 'T 1 l 1 l l Cl. l _.._"". _..._... _... _..._.._..._.,_..._..._. II J:; " : ;......,,, ) _. "., N 1 a l

32 ... '_.... <.... _,.... _. '._'. _._... _,_.._....,_c_ _ rl 1'1') Lower_ ail crit. poin ,'_,_... _..,... _ _._ p =.95 p =.95 Lower tail probability level a..._'..,,..._...,_._.._._.._._.._._._..._""_...._...,. "....._. b'. _ _.._._.. _. Lower tail probability level a _.. _._. '"''...._..._.._._ '_.'","'_.,, _... _..._._._._. N a N 1 1 (l1 N l'j 1 (l1 (l1 H' (l1 TIT (l1 1'1 l'j 1 '1 "1 1 a a ,_..._._., w.._ _..'. _ ",. "...._ _ l l oln t a<" O!l ""," ,.'" '

33 ..._...._ N I") _. Lower _... tail critical points..... w _.....,"_._...._......_._ _ _, '., _, p =.95 p =.95 Lower tail probability level (l _.._.._...._ Lower tail probability level a.._ _.. _...._...,_. _......_..._.'.., K X rr X _._.._._..."_.,...' _.. N l (l1 N 1 (l1 N l (l1 1\1 1 a 1 N 1 a 1 "1,.,..oO_.,'. =,..._..... _...,... _....._ '. (l1 N 1 a 1 N 1 _.._ (l l ::" : : ,'" " , , , , _., " ; ,, _

34 .. _.." ti') ti') Upper tail critical points p =.95 p =.95 Upper tail probability level a, Upper tail probability level a 4 4 N a TeT a N 2 2 a N a '"2 2!J a N 2 2 C!2 JlT 2 lx N '2 a 2 " ll 9 a ". O'.. _ ,,, =,, o 'J ' _ , ",' '., _.. 0 0'

35 _.. _._. f") Upper tail critica:j._ ;ps>ints p =.95 p =.95 Upper tail probability level ex Upper tail probability level a _.. N 2 a 2 N 2 a 2 N 2 a 2 J'i 2 a 2! l\t 2 a 2 n N 2 a _ ; OO r,.' ).087 &..., = " _ _ v oB

36 9 Lfl rj") Upper tail critical poip p =.95 Upper tail probability level a K X N N N N 0.2 ;:> , _ ' \ olo

37 R._...._..,'._. _.._._._...._..._.._._.._..._..._._._... _.,._.. 58 ", \0 I"').._...._.._._, _.._._._. Lower tail critical points,... _._._._,._..._._..._.. p = 99 P =.99 _'... "..._. J_...."',_ Lower tail probability level Lower tail probability level... _.,_... _ _ _....._.... _.w._. K X I( v f _.,.... _ _.._ _,..._ N ' N l 1 N l 1 N l 1 "'r 1 N 'J (1,1 N 1 ''''1 l C1. 1,_.. _ _... '...'_._ () 3 ' 'i """ ) , : 'I ooh ooa i <:, <

TABLES OF DISTRIBUTIONS OF QUADRATIC FORMS IN CENTRAL NORMAL VARIABLES II. Institute of Statistics Mimeo Series No. 557.

TABLES OF DISTRIBUTIONS OF QUADRATIC FORMS IN CENTRAL NORMAL VARIABLES II N. L. Johnson* and Samuel Kotz** University of North Carolina Chapel Hill, N. C. Temple University Philadelphia, Pa. Institute