Mitwalli, Saleh M.

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Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaSaleh M. MitwalliAn occupancy problem with group drawings of different sizesMathematica Slovaca, Vol. 52 (2002), No. 2, 235--242Persistent URL: http://dml.cz/dmlcz/132830Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2002Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©2002 . - i L r . - . ^ /~r\r\r*\ M ~ «or- o*>-. Mathematical Institute Math. SlOVaCa, 52 (2002), NO. 2, 235-242 Slovak Academy of Sciences AN OCCUPANCY PROBLEM WITH GROUP DRAWINGS OF DIFFERENT SIZES SALEH M. MITWALLI (Communicated by Gejza Wimmer) ABSTRACT. We consider the classical occupancy problem with group drawings of different sizes. We derive the exact distribution of the number of balls occurring in all sampled groups and its asymptotic behavior. Furthermore , the numbe r of drawings until at most k balls are observed in all samples is studied. 1. Introduction Consider an urn containing s different balls. Groups of mi balls (mi < s, m{ ^ m2 y£ •••) are drawn successively with replacement; in each draw-ing all (jn ) , i = 1 ,2 , . . . , possible samples of balls are equally likely. Let U-(-s,7t7l,7n2,... ,7nn ,n) be the number of balls occurring exactly j times in the first 77 drawings. Problems concerning the distribution of {UAs,ml,m2,... . . . , miV n) : j = 0 , 1 , . . . , n , n G N} or of functions of this sequence are called occupancy problems. Their history can be traced back to DeMoivre, Laplace, and Euler (see S t a d j e (1990)). For the case m1 = m2 = • • • = mn = 1 we refer to H o i s t (1986) for a survey and a unifying approach. For arbi-trary m1 = m2 = •••=-: rnn = I > 1 in particular U0(,s,/,n), the number of balls that have not shown up in n drawings, has been extensively stud-ied. H a r r i s et al. (1987) derive its finite and asymptotic distribution (in their model each sampled ball has a probability q G [0,1] of "disappear-ing" after being drawn). Asymptotic results on U0(s,l:n) can be found e.g. in M i k h a i l o v (1980) and P a r k (1981). If s = 2/, the exact probability distributions of C/0(«s,/,n) and Un(s,l,n) are equal. In this note we consider C/(s, m[,m2,..., ran, n) = Un(s, m1? ra2,..., mn), the number of balls occurring in all n sampled groups, which is a generalization of S t a d j e et al. (1998). We determine its exact distribution and its asymptotic behavior for fixed 7t, as 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Primary 60E05, 62E15. K e y w o r d s : occupancy problem, waiting times. 23Г SALEH M. MITWALLI s,mvm2,... , m n —> oo. If ^-, m = ml = m2 = • • • = mn , is fixed, U(s,m, n) n is asymptotically normal, while for fixed s fj ̂  it tends to a Poisson distribu-i=l tion. The functional limit theorem for exchangeable random variables and the method of moments are used in the proofs. 2. The exact distribution We derive formulas for the probability distribution of U(s, m1, m 2 , . . . , m , n) b-i and its factorial moments. Let (a)b = fj (a — i) for a,b £ Z + and /iA = i=o E((f / (5, m2 , m 2 , . . . , m n , n))AJ . T H E O R E M (1). We have P(U(s,mvm2,... ,mn,n) = Л) m i n { 7 7 г i , . . . , 7 7 i n } — A, =0 £ (-i)-(';*)п «=o \ г / / = 1 / s—k—i \ \irn—k — iJ (') A; = 0 , l , . . . , m m { m 1 , m 2 , . . . , m n } , H = (s)к П K ) A řЛ V (*)* , ( І ) (2) ;'j iч (3) P r o o f . We use the inclusion-exclusion formula for the events B in all n sampled groups", j = 1, 2 , . . . , 5. We obtain P(exactly k of Bx,..., Bs occur) = £<-i)f ̂  £ P(fl,,n...nBJ.J. »=0 V ' l<ji<--<jk+i<s Since the n drawings are independent, f ft [( *'',)/( s )] if l€ { 0 , . . . , m i n { n . 1 , m „ . . . , 7 n }} P ( 5 J . i n - - - n B i l ) = < ,= 1 lKm'-"'Km>'\ x ' ' x v 2 ' ' » / ; ! 0 if l > m i n { m 1 , m 2 , . . . , mn} . It follows from (3) and (4) that P(U(s,mvm2,...,mn,n) - k) m i n { 7 7 i i , . . . , m „ } A; , x 7 \ ?* - E <-<r G;, n 7 0 X 7 V 7 / I (4) C, 23G AN OCCUPANCY PROBLEM WITH GROUP DRAWINGS OF DIFFERENT SIZES which, after a slight simplification, yields (1). To prove (2), note that the conditional distribution of U(s, m1,m2,..., mn, n) given U(s,ml,m2,... ,mn_i,n — 1) is hypergeometric Setting U(n) = U(s,ml,m2,...,m,n,n) we have >(U(n) = k\U(n-l)=j) = fy i\ ( S~j ) J \\mn—k' (5) ( ") ' \m„/ Thus, v({U{n))k | U(n - 1) = j) = (j)k(mn)k/(s)k (6) (see J o h n s o n and K o t z [4; p. 81]) and, by (6), E((tf (»))*) = [(mn)k/(s)k]E((U(n-l))k) = fl[(mi)J(s)k]E({U(l))h) i=2 71 ПtЮdШЮfc-i=2 3. Asymptotic results For obtaining asymptotic results on U(s,m1,m2,... ,mn,n) the following theorem is crucial. THEOREM (2). Let s —•> co and m- —> oo for all i = l,...,n such that X{= lim rs± exists and X{e (0,1). Let ln= f[ A-2( £ ( l - A - " 1 ) - l ) + f[ A-. s-+°° t = l V i = l ' i=l Then 7n = ton s~lVa.r(U(s,m1,m2,... ,mn,n)) (7) and for every e > 0, 51 £ls 1U(s,ml,m2,...,mn,n)-Y[(mi/s) j —rO ^ i = i ' m L2 , as s —> oo . (8) P r o o f . It follows from (2) that n E(U(s, mvm2, ...,mn,n))=s J\(mjs) (9) i=\ 237 SALEH M . MITWALLI and Var(/7 ( s ,m 1 ,m 2 , . . . , m n , n ) ) n JI n li m ť ( m ť - 1) ~mi ]7 ™2 i=\ + І=\ 7 = 1 (10) 5 n - Ҷ ő _ 1 )n-l 5 n - l ő2n-2 n / ч 2 n л — 1 n n / ч 2" _ І = 1 Ч 7 ѓ = l 7 = 1 i = l x 7 _ Since ѓ = l ч v 1 — m] _ - i 1 Ьй-A-1), i=l equation (10) yields S-1Var(U(5,m1,m2,...,mn,n))^r[VE(1-Ar1)-nA" + nA<' П П ѓ = l ѓ = l i=\ 7 = 1 proving (7). Assertion (8) now follows from E( \s 1U(s,mi,i l l K W l 2 ) = V a r ^ C t ^ m . , 7 = 1 J ' m 2 , . . . , m n , n ) ) s 7 n , as s —> oo. D THEOREM (3). Under the conditions in Theorem (2). if A = A: = • • • = An , let m\ = m2 = • • • = ™>n = m = m(s) as function of s and U(s, mX)m2,..., mn n = U(s, m, n ) . T/ie asymptotic behavior of U(s, m, n) is s~~l/2(U(s,m,n) - (mn/sn~1)) —> N(0,an) in distribution, as where an = A2 n ( n ( l - A" 1 ) - l) + An = lim s - 1 V a r ( U ( s , m , n ) ) . s — > OC (11 P r o o f . First consider the case n = 2. Let (X[& , . . . , X^) be an equidis-tributed random permutation of the numbers 1,. . . , s. A short reflection shows that the random variable V , = # { t e { l , . . . , m } : X(ts) <m} has the same distribution of U(s,m,2). Define the number a^(i) for s G N and i = 1, . . . , s by a(s}(i) [(s — m)/ms] , i 1,. . , m , ,1/2 [ÎІI/(S(S — m))] , i m + 1,. . , s 238 AN OCCUPANCY PROBLEM WITH GROUP DRAWINGS OF DIFFERENT SIZES and introduce the D[0,1]-valued process V ( s ) = (V ( s )(i)) /6[0 xl by setting [st] V<*>(t) = 5 > ( ' ) ( . Y ) ' ) ) . *€[0,1] . i=l Since and J2ais)(i) = 0, 2>W(i))2 = l ?:=i 7 = 1 max |a^(i) l —> 0 as s —> co , Ki<s • W | we can use a functional limit theorem for exchangeable random variables (see A 1 d o u s (1985)) and conclude that V^5^ converges in distribution to a Brown-ian bridge W° = (W°(«)) te[0 i r This yields V w ( f ) ^ W ° ( A ) as s—>oo. (12) But a short computation shows that V("(f) = (^)1 / 2VS - (-(-?-u)l/2(m-V.) = (v .—(?) a ) / (? ( i -? )«) 1 / 2 (13) g{u(s,m,n)-s(f)2)/{f(l-^)s)] 1/2 As W°(A) is 7V(0, A(l - A)) -distributed, m/s —> A , and a2 = A2(l - A)2, wTe obtain assertion (11) for n = 2 from (13) and (12). We can now proceed by complete induction. Suppose that the theorem holds for some n > 2. Then let Ys be a random variable that has the same distribution as U(s, m, 77) and is independent of (Ar{ , . . . , X^) and thus also of the process (V(s)(£))/(_[0 x]. Then as V( s ) A W° , the induction hypothesis implies that ( V ( s , ( 0 + ( ^ ^ y ) 1 / 2 [ Y , - S ( m / S ) » ] ) / e [ M A ( W ( t ) + Z) t 6 [ 0 i ] ] 1 (14) where Z is jV(0, -rzy^J -distributed and independent of W° . Let us take for t in (14) the random time t = Ya/.s. By Theorem (2), YJ.s —•» An in L2 so that V('s) ( ^ ) + ( ^ ) V 2 [Y, - s(m/s)n] A W°(A") + Z . (15) 239 SALEH M. MITWALLI The limiting distribution in (15) is JV(0, A"(l - A")) * N(0, ^ a n ) = -V(0, -g±fr) To evaluate the left-hand side of (15), let V3 = #{z: 1<*<YS , XIS)<YS}. Then v (ä)(^)=É« (s)Hs)) ѓ = l = * s\ sm ) ~~ l*s ~ * J \s{s-m) ) i=ì ll2 , ~ s / Ч 1 / 2 so that V ( S )(^) + ( Ä ) > . - ( ? П ~V.sП~.~r7Гj + ^5(s-m)J ) ~ (S(s-77г)J 5 ( ~ ? ) _*-1/2(v.-s(?)n+1) (16) ( T ( I - T ) ) 1 / 2 Now note that by (5), V s = # { i : l < i < Y s , x |s ) < Y , } = C/(S,m,n + l ) . (17) Combining (15), (16) and (17) we obtain (11) for n+1 instead of n. The theorem is proved. D n THEOREM (4). Assume that s f_ (2jL) —> a as s —> oo, for some a > 0 z = l and some n G N. Then c7(«s,m_,m2,... ,mlVn) converges to the Poisson distribution na with mean a. P r o o f . By (2), the factorial moments fik of U(s, m_, m 2 , . . . , mn, n) con­verge to the corresponding factorial moments ak of 7rtt. Thus, the moments of U(s, m1,m2,..., m n , n) also converge to the corresponding moments of 7rtt . Since 7rtt is the only distribution on R with this moment sequence, U(s, /??_, m 2 , . . . , m n , n) must converge to 7rtt in distribution. D 240 AN OCCUPANCY PROBLEM WITH GROUP DRAWINGS OF DIFFERENT SIZES 4. The waiting times The number of balls occurring in all n sampled groups U(s,m1,m2,... . . . , m n , n ) , n = 1,2,... , may be considered as a system observed at time points n. The system will be said to be in state i if and only if U(s,ml,m2,..., m n , n) = i, i = 0 , 1 , . . . , min{m1, ra2,..., ran}. The transition probabilities are p . ^ n ) = P\U(s,mvm2,... ,mn,n) =j \ U(s,mvm2,... ,mn_x,n - 1) = ij Vj/ \mn-jJ \mnJ and the initial distribution P(U(s,m1,m2,... , ra n ,0) = mx) = 1, and P(U(s,m1,m2,... , ra n ,0) = i) = 0, i = 0 , 1 , . . . ,ml — 1. This completely defines a discrete nonhomogeneous Markov chain with states 0 , 1 , . . . , min{ra ], ra2,... . . . , m n } . We denote by Tk (the waiting time) the random variable represent­ing the number of drawings necessary, after which at most k elements will be observed in all drawings. The probability distribution of Tk is P(r f c = n + 1 ) 7?i / \ n — 1 / \ —1 m — h / j v = E : n ; E(-')"/-h=k+l X 7 1=1 X l / i=0 V ' j = 0, l,...,i, П Í í ' л ' i ì í i - í í ì " ï , ' f c ' i ì 7=i \mj~h~*/ [ Vmn/ \mn-Һ-ІJ where the random variables Tk and U(s, ml,m2,..., mn, n) are related to each other in an obvious way for the distribution function P(rA. > n) = P(U(s,ml,m2,... , r a n , n ) > fc) and, by (1), the result follows. The absorbtion t ime into state zero. Denote by r the first entrance time into state zero, i.e., r = min{n : U(s,m1,m,2,... , r a n , n ) = 0} . Then P ( т = n) = $ > - ! ) ' 3 = 1 n-ì п С4) \mr—jJ r = l V771.,/ \mn-jJ __ 1 (•') using the facts that P ( r = n) = P ( r < n ) - P ( r < n- 1) = P ( { 7 ( s , m 1 , r a 2 , . . . , r a n , n ) = 0) - P(U(s,mvm2,..., m n , n - l ) = 0) 241 SALEH M . MITWALLI and P{U(s,mvm2,...,mn,n) = 0) = ү/(-lýГ j = o n í s-j \ n \mr-jJ ( s ) r=\ \mnJ the result follows. Also, the pth factorial moments of r are P-1 ( »-]' \ ra ( - 1 ) \mr — iJ E((r)p)=p.nfíf D-Dj+i E n fn r=\ \mr) j = \ ^J/ n=0 i=p U J oo n + p - 1 / s-j Í*-J\ \71li—jj r ( s~j ) \mp+n-jJ ( s ) - i REFERENCES [1] ALDOUS, D. J . : Exchangeability and related topics. In: Ecole d 'Ete de Probabilites de Saint-Flour XIII - 1983 (P. L. Hennequin, ed.), Lecture Notes in Math. 1117, Springer, New York, 1985, pp. 1-199. [2] HARRIS, B.- MARDEN, M.—PARK, C J . : The distribution of the number of emptj cells in a generalized random allocation scheme. In: Random Graphs '85 (M. Karonski, Z. Palka, eds.), Ann. Discrete Math. 33, North-Holland, Amsterdam, 1987, pp. 77-90. [3] HOLSТ, L.: On birthday, collector's, occupance, committee and other classical urn prob-lems, Internat . Statist . Rev. 54 (1986), 15-27. [4] JOHNSON, N. L — К O Т Z , S.: Urn Models and Their Application, Wiley, New York, 1977. [5] KOLCHIN, V. F — SEVASТYANOV, B. A — CHISТYAKOV, V. P . : Bandom Allocations, Winston, Washington D . C , 1978. [6] MIKHAILOV, V. G . : Asymptotic normality of the number of empty cells for group allo-cation of particles, Тheory Probab. Appl. 25 (1980), 82 90. [7] PARK, C J . : On the distribution of the number of unobserved elements when m-samples of size n are drawn from a finite population, Comm. Statist. Тheory Methods 10 (1981), 371 383. [8] S Т A D J E , W . : The Collector's problem with group drawings, Adv. Appl. Probab. 22 (1990), 866-882. [9] S Т A D J E , W . — M I Т W A L L I , S.: An occupancy problem with group drawings I. In: Тһe 23rd International Conference for Statistics, Computer Science, and its Applications, Cairo, Egypt, May 9-14, 1998. Received December 28, 2000 Revised June 25, 2001 Department of Mathematics Faculty of Science at Damiatta Mansoura University P.O. 34517(3) New Damiatta EGYPT 242 

An occupancy problem with group drawings of different sizes

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An occupancy problem with group drawings of different sizes

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Institute of Mathematics AS CR, v. v. i.