DISTRIBUTIONS OF NUMBERS OF FAILURES AND SUCCESSES UNTIL THE FIRST CONSECUTIVE k SUCCESSES*

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1 Ann. Inst. Statist. Math. Vol. 46, No. 1, (1994) DISTRIBUTIONS OF NUMBERS OF FAILURES AND SUCCESSES UNTIL THE FIRST CONSECUTIVE SUCCESSES* i Department SIGEO AKI 1 AND KATUOMI HIRANO 2 of Mathematical Science, Faculty of Engineering Science, Osaa University, Machianeyama-eho, Toyonaa, Osaa 560, Japan 2 The Institute of Statistical Mathematics, ~{-6-7 Minami-Azabu, Minato-u, Toyo 106, Japan (Received March 15, 1993; revised June 21, 1993) Abstract. Exact distributions of the numbers of failures, successes and successes with indices no less than l (1 < l < - 1) until the first consecutive successes are obtained for some {0, 1}-valued random sequences such as a sequence of independent and identically distributed (iid) trials, a homogeneous Marov chain and a binary sequence of order. The number of failures until the first consecutive successes follows the geometric distribution with an appropriate parameter for each of the above three cases. When the {0, 1}- sequence is an iid sequence or a Marov chain, the distribution of the number of successes with indices no less than l is shown to be a shifted geometric distribution of order - l. When the {0, 1}-sequence is a binary sequence of order, the corresponding number follows a shifted version of an extended geometric distribution of order - l. Key words and phrases: Geometric distribution, discrete distributions, Marov chain, waiting time, geometric distribution of order, iid sequence, binary sequence of order, inverse sampling. 1. Introduction Let Xl, X2,... be a sequence of {0, 1}-valued random variables. We often call Xn the n-th trial and we say success and failure for the outcomes "1" and "0", respectively. The distribution of the number of trials until the first consecutive successes is well nown when the sequence is an iid sequence with success probability p (e.g. cf. Feller (1968)). The probability generating function (pgf) of it is given by p(1-1 - t + qp~t +l ' * This research was partially supported by the ISM Cooperative Research Program (92-ISM- CRP-16) of the Institute of Statistical Mathematics. 193

194 SIGEO AKI AND KATUOMI HIRANO where q = 1 -p. Philippou et al. (1983) called it the geometric distribution of order and derived its exact probability function.

2 194 SIGEO AKI AND KATUOMI HIRANO where q = 1 -p. Philippou et al. (1983) called it the geometric distribution of order and derived its exact probability function. Since then exact distribution theory for so called discrete distributions of order has been extensively developed and some of the results were applied to problems on the reliability of the consecutive--out-of-n:f system by many authors (cf. Ai et al. (1984), Ai (1985, 1992), Charalambides (1986), Hirano (1986), Philippou (1986, 1988), Ling (1988), Ai and Hirano (1989, 1993), Ebneshahrashoob and Sobe] (1990), Sobel and Ebneshahrashoob (1992) and references therein). Every distribution of order 1 is of course the usual corresponding discrete distribution. This is an explicit relation between distributions of order and that of order 1. However, as far as we now, no other relations among distributions of different orders have been studied. In this paper, we find out a vector of random variables whose marginal distributions follow the geometric or the extended geometric (eft Ai (1985) or Section 3) distributions of order l (1 < 1 < - 1). This leads us a new type of geneses of the geometric distributions of order I. Surprisingly, this property only depends on the conditional probability P(X~+I = 1 I X~ = 1) and hence we can treat homogeneous {0, 1}-valued Marov chain similarly as the iid sequence. Throughout the paper let and l be fixed positive integers such that l < - 1. We denote by G(p) the geometric distribution of order, by G(p, a) the shifted geometric distribution of order so that its support begins with a, and by G(p, a; x) the probability of G~(p, a) at x (x = a, a + 1, a + 2,...). By definition, G(p) = G(p,) holds for every > 1. Here the geometric distribution, to be denoted by G(p), is defined as the distribution of the number of failures preceding the first success. Note that G(p) = GI(p, 0). In Section 2, only an iid sequence is treated and our main idea is illustrated. Exact distributions of the numbers of occurrences of "0", "1" and "1" with indices no less than 1 until the first consecutive successes are shown to be G(p), G-l(p, ) and G-t(p, ), respectively. In Section 3, results given in Section 2 are extended for dependent sequences such as the homogeneous Marov chain and the binary sequences of order. When the {0, 1}-sequence follows the Marov chain, the distribution of the number of occurrences of "1" with indices no less than l until the first consecutive successes becomes G-l(pll, ) where Pll = P(Xi+I = 1 I Xi = 1) for i = 0, 1, 2,... When the {0, 1}-sequence is a binary sequence of order with parameters pl,..., P, the distribution of the number of occurrences of "1" with indices no less than 1 until the first consecutive successes is shown to be the shifted extended geometric distribution of order - l with parameters pt+l,...,p whose support begins with - l Independent trials 2.1 Number of failures until the first consecutive successes Let X1,X2,... be a sequence of lid {0, 1}-valued random variables with P(X1 = 1) = p (success probability). We denote by T the waiting time (the number of trials) for the first consecutive successes. Let ~ be the number of occurrences of "0" (failure) among X1,X2,...,X~. For nonnegative integers rn

3 NUMBERS OF FAILURES AND SUCCESSES 195 and N, let d(m, N) - P(u = m, ~- = N). Then we have d(m,n)=o d(0, ) = p if0<_n<, d(m,)=o ifm 0, (2.1) d(m,n) =0 ifn> andre< 1, --1 d(.~, N) : Z p~qd('~ - 1, x - l - 1). /=0 The last recurrence relation in (2.1) follows immediately if we consider all possibilities of the first occurrence of "0". Let (s, t) be the joint pgf of (u, ~-), i.e., ~(s,t) = ~ m=0 N=0 d(.~,n)s~t N The relations (2.1) imply PROPOSITION 2.1. The Pgf (s,t) is given by pt~,(s,t) = 1 - E~o 1 p~qst~+l COROLLARY 2.1. The marginal distribution of u is the geometric distribution with parameter p, i.e., u ~ G(p). We denote by t771 and E2 a "1"-run of length and a "0"-run of length r, respectively. Let u 0 be the number of occurrences of E2 among X1, X2,..., X~. Let P2(U) be the probability of the event that at the u-th trial the sooner event between E1 and E2 occurs and the sooner event is E2. We set and Then, we can obtain (2.2) OO (t) = Zp2(u)t~ U:T d~ (.~, N) = P(.o =.~, ~- = X). / dl(m,n)=o d~(m,)=o d~ (0, ) = p ifo_<n<, ifm O, N- dx(.~, x) = Z :D~(~)d~(~- 1,X - ~).

4 196 SIGEO AKI AND KATUOMI HIRANO Let 1 (s, t) be the joint pgf of (Uo, T), i.e., Prom the relations (2.2) we have l(s,t)= fi fi dl(m,n)s'~t N. m=o N=O PROPOSITION 2.2. The pgf l(s,t) is given by l(s, t) = EN:0 dl(0, N)t N 1 - s (t) COROLLARY 2.2. The marginal distribution of uo is the geometric distribution with parameter 1 - ~(1), i.e., i - I) - ] _ se(1)' wherc (2.3) qr-l(1 _p) (1) = 1 -- (1 -p-1)(l - qr-1)' Remar. By definition, (1) = P({E2 comes sooner}). Hence, the formula (2.3) can be obtained anyway. But, one of the easiest way to find out it is to set x = 0, y = 1 and t = 1 in the generalized pgf (2.2) in Ebneshahrashoob and Sobel (1990). We can give an intuitive explanation why ~0 has the geometric distribution. We cut the {0, 1}-sequence at the trial when the event E1 or E2 has just occurred. Then every divided part of the sequence has the distribution of the waiting time for the sooner event between E1 and E2. Hence, we have P({u0 = m}) = {P({E2 comes sooner})}'~p({e1 comes sooner}). 2.2 Number of successes until the first consecutive successes In this subsection we consider the distribution of the number of successes until the first occurrence of consecutive successes. We denote by ul the number of occurrences of "1" among X1, X2,..., X~. Set A(x) = P(ul = x). PROPOSITION 2.3. The probabilities {A(x)}~_ 0 satisfy the relations (2.4) A(x) = 0 ifo<x<, A() p-1 A(x) qd(x) +pqa(x- 1) p-xqa(x - + 1) ifx >.

5 NUMBERS OF FAILURES AND SUCCESSES 197 PROOF. Note that {/Jl = } = Ua _0{O"" }. Then, OO A() = ~ j=0 qjp = p-1. By considering all possibilities of the first occurrence of "0", we easily obtain the above recurrence relations. [] Let 1(t) be the pgf of zjl, i.e., 1(t) = ~x~ 0 A(x)t x. From (2.4) we immediately obtain PROPOSITION 2.4. The Pgf i(~) is given by p-x (1 - pt)t 1(t) = i 7tT~ ~tt" Remar. Proposition 2.4 shows that z~l follows the shifted geometric distribution of order - 1 so that the support begins with, i.e., ~1 ~ G-l(p, ). Now, in order to derive a general result, we define the following "words" or "sets of words" in {0, 1}-sequences as follows: Let S = "1", F1 = "0" and 5cl - {F1}. We define a set of words Y2 - u j_ i{0.--0x} ~ (= We say "F 2 occurs" when one of words in J j 52 occurs in the {0, 1}-sequence. It is easy to see that any {0, 1}-sequence which ends with "1..- 1" can be uniquely regarded as {S, F2}-sequence which ends with "S...S".~ Next, we consider a set of words 5-a --- U~_I{~S} in {S, F2}- -x j sequences which ends with "S...S". We say "F3 occurs" when one of words -1 in 53 occurs in the {S, F2}-sequence. Then, any {0, 1}-sequence which ends with "1.-.~1" can be uniquely regarded as {S, Fa}-sequence which ends with S~2_S. By -2 repeating these considerations, we can define 51,..., 5o and F1,..., F. Thus, for i = 1, 2,...,, {0, 1}-random sequence which ends with "1... 1" can be regarded as the new binary {S, Fi}-sequence which ends with "S.--S". --i+1 PROPOSITION 2.5. For i = 1,...,, P(Fi) = q, which doe8 not depend on i. PROOF. For i = 1, P(F1) = P({O}) = q. Assume that P(Fi) = q holds. Then we have P(Fi+I) = ~-~oo j=i(p(fi))jp(s) = }-'.j=l oo qjp = q. This completes the proof by the induction. []

6 198 SIGEO AKI AND KATUOMI HIRANO Now, we denote by Wl the waiting time for the first occurrence of "1... 1" Also W1 -= 0. In a {0, 1}-sequence, fix a success "1". We say that the "1" is a 'T' (success) with indices m when there exists a "1"-run of length m - 1 just before the "1". As an example we give a {0, 1}-sequence "0, 11,0, 11, 12, 0, 11, 12, 13, 14, 15, 0, 11", where every subscript of "1" shows its indices. Fix an integer 1. Let ul be the number of l's with indices no less than l until the first consecutive successes. Note that uz = the number of occurrences of "1" with indices no less than l between Wl and W+l. Remar. The number of "l's" with indices no less than l coincides with the number of success runs of length l by the overlapping way of counting (el. Ling (198S)). THEOREM 2.1. The distribution of ut is the shifted geometric distribution of order - l, i.e., ~l ~ G-l(p, ). PROOF. Note that = x) = P(w = = x I w, = W For the {0, 1}-sequence, regard the part just after Wz as the binary {S, Fz}- sequence. Then, the number of occurrences of S until the first occurrence of a S-run of length ( ) is equal to ut. From Proposition 2.4, the distribution is the shifted geometric distribution of order - l and does not depend on the value w. Thus we have P(uz = x) = a-l(p, - l + 1; x). This completes the proof. [] 3. Dependent sequences 3.1 Marov chain In this section we study the problems in the previous section based on two typical dependent sequences. One is a two-state homogeneous Marov chain and the other is a binary sequence of order defined by Ai (1985). First, we consider the problems based on the following Marov chain. Suppose that the sequence Xo, X1,X2,... is a {0, 1}-valued Marov chain with P(Xo = O) = Po, P(Xo = 1) =- Pl, and for i = O, 1, 2,... P(Xi+l = 0 ] X~ --- O) -- Poo, P(Xi+l = 1 I = o) = pol, P(Xi+l = o I = 1) = plo and P(Xi+ 1 = 1 I Xi ~- 1) = P11. For dependent sequences, we use the same notations as the iid case. The distribution of the number of trials until the first consecutive successes in the Marov chain was obtained by Ai and Hirano (1993).

7 NUMBERS OF FAILURES AND SUCCESSES 199 THEOREM 3.1. Let ~ be the number of failures until the first consecutive successes in the Marov chain. Then the conditional distribution of ~, given that Xo = 0 is the geometric distribution G(polp~). The conditional distribution of L, given that Xo = 1 is the mizture distribution Pll. 5o + (1 -pll ) Gl(POlPl~ 1, 1), where 5o means the unit mass at the origin. PROOF. First, we consider the case that X0 = 0. Define a word and a set of words by S. "1.. 1". and. jr U. -1 o{ }.. We. say F occurs" when 3 = j a word in jc occurs. If we regard the {0, 1}-sequence until the first "1-.- 1" as the new binary {S, F}-sequence, then the new sequence becomes a Marov chain. Since we consider the {0, 1}-sequence only up to the first "1... 1" the all possible sequences we can observe are {FF---F S}j%0. Then, from the Marov property of J the sequences and the assumption that X0 = 0, we have P(FF... FS) = {P(F I Y J F)}JP(S I F). This probability law is the same as the iid trials with success probability polp~ 1. In the iid sequence, the number of occurrences of F until the first S is equal to L,, since every word in 5 has just one "0". Thus the conditional distribution given that X0 = 0 is the geometric distribution G(PolPlll). Next, we consider the case that Xo = 1. If a "0" occurs before the first " " then the distribution of the sequence just after the first "0" is the same as the above case. The probability that "0" does not occur until the first "1--- 1" is Pll. Hence, we have the desired result. [] THEOREM 3.2. Let 111 be the number of successes until the first consecutive successes in the Marov chain. Then Yl follows the shifted geometric distribution of order - 1, i.e., ~'l ~ G-l(p11, ). PROOF. Since the case that = 1 is trivial, we assume that > 1. Let W2 be the waiting time for the first "1" and let W+l be the waiting time for the first "1... 1". Note that W2 < W+l with probability one. Define a word and a set of words by S = "1" and )c2 = U~_l{"0...01"}. We say that "F2 occurs" when a J word in -7"2 occurs. We regard the {0, 1}-sequence just after W2 as the new binary {S, F2}-sequence. Noting that c~ P(r2) = Zpl0p 0pol = pl0 and P(S) = p l, j=0 we see that the new {S, F2}-sequence is the iid trials with success probablity pll. The event that "1... 1" occurs for the first time for the {0, 1}-sequence just after

8 200 SIGEO AKI AND KATUOMI HIRANO W2 is equivalent to the event that "S... S" occurs for the first time in the new --1 {S, F2}-sequence. Since W2 < W+l holds with probability one, we see that Ul = 1 + (the number of trials until the first consecutive - 1 successes in iid trials with success probability Pn). This completes the proof. [] Remar. The iid sequence treated in Section 2 is the special case of the Marov chain (Pl = Pl1 = P01 -- P). So, corresponding results in Section 2 can be regarded as corollaries of those in this section. From the proof of Theorem 3.2, the {0, 1}-sequence just after the first "1" can be regarded as Bernoulli trials with success probability P11. Then Theorem 2.1 immediately implies THEOREM 3.3. For l = 1,..., - 1, let r'l be the number of successes with indices no less than 1 in the Marov chain. Then the distribution of ~l is the shifted geometric distribution of order - l, i.e., ~z ~ G-z(p,l, - l + 1). 3.2 Binary sequence of order A binary sequence of order is defined and studied by Ai (1985, 1992). This is an extension of iid trials and determined by parameters Pl,...,P. The definition is as follows: DEFINITION. A sequence {Xi}~=0 of {0, 1}-valued random variables is said to be a binary sequence of order if there exist a positive integer and real numbers 0 < Pl,P2,..., P < 1 such that (1) X0 = 0 almost surely, and (2) P(X~ = 1 I Xo = xo, X1 = xl,...,x~_i = x~-l) = pj is satisfied for any positive integer n, where j = r - [(r - 1)/], and r is the smallest positive integer which satisfies x~_~ = 0. The distribution of the number of trials until the first occurrence of consecutive successes is the extended geometric distribution of order, whose pgf is given by Pl Pt 1 - )-~'~i=1pl "Pi-lqit ~ where q~ = i -Pi (cf. Ai (1985)). We denote by EG(pl,...,p, a) the shifted extended geometric distribution of order so that its suport begins with a. THEOREM 3.4. Let ~'o be the number of failures until the first occurrence of consecutive successes in the binary sequence of order. Then, L'o follows the geometric distribution G(pl.. P ). PROOF. If we define S and F samely as in the proof of Theorem 3.1, then the {S, F}-sequence obtained from the {0, 1}-sequence becomes lid trials with success probability Pl " P. Since every F includes just one "0", the number ~0 coincides

9 NUMBERS OF FAILURES AND SUCCESSES 201 with the number of occurrences of F until the first S. Hence the distribution of u0 is the geometric distribution, G(pl'"p). [] THEOREM 3.5. Let ~1 be the number of successes until the first consecutive successes in the binary sequence of order. Then ~1 follows the shifted extended geometric distribution of order - 1, i.e., ~1 ~ EG-I (p2,...,p, ). PROOF. Since the case that = 1 is trivial, we assume that > 1. Let W2 be the waiting time for the first "1" and let W+l be the waiting time for the first "i --. i". Note that W2 < W+i with probability one. Define a word and a set of words by S = "1" and Jr2 = tj~= 1{ "0 0 1" }. We say that "['2 occurs" J when a word in Jr2 occurs. We regard the {0, 1}-sequence just after W2 as the new binary {S, F2}-sequence. Then the {S, F2}-sequence becomes the binary sequence of order - 1 with parameters P2,...,P. Since both S and every word F2 E ~c2 include just one "1", we see that Pl = 1 -~- (the number of trials until the first occurrence of a S-run of length - 1 in the binary sequence of order - 1). This completes the proof. [] THEOREM 3.6. For l = 1,..., - 1, let Yl be the number of successes with indices no less than l in the binary sequence of order. Then the distribution of ~l is the shifted extended geometric distribution of order - l, i.e., ~z EG-z(pl+l,...,p, ). PROOF. When 1 = 1, the statement is reduced to Theorem 3.5. So, we fix an integer 1 (2 < I < ). Define W1,..., W and W+l similarly as in Section 2. Then, we can see that the number ~l is equal to the number of occurrences of "1" with indices no less than 1 between W1 and W~+I. If we regard the {0, 1}- sequence just after W1 as {S, Fl}-sequence, then the sequence becomes the binary sequence of order ( ) with parameters Pl,..., P. Noting that the number of occurrences of "1" with indices no less than I between Wl and W+l coincides with the number of occurrences of S in the {S, Fz}-sequence until the first occurrence of a S-run of length ( - l + 1). Then, Theorem 3.5 implies that the distribution of ~z is EG-i(Pl+i,...,P, ). This completes the proof. [] REFERENCES Ai, S. (1985). Discrete distributions of order on a binary sequence, Ann. Inst. Statist. Math., 37, Ai, S. (1992). Waiting time problems for a sequence of discrete random variables, Ann. Inst. Statist. Math., 44, Ai, S. and Hirano, K. (1989). Estimation of parameters in the discrete distributions of order, Ann. Inst. Statist. Math., 41, Ai, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Marov chain, Statistical Sciences and Data Analysis; Proceedings of the Third Pacific Area Statistical Conference (eds. K. Matusita, M. L. Puri and T. Hayaawa), , VSP International Science Publishers, Zeist.

202 SIGEO AKI AND KATUOMI HIRANO Ai, S., Kuboi, H. and Hirano, K. (1984). On discrete distributions of order, Ann. Inst. Statist. Math., 36, 431-440. Charalambides, Ch. A. (1986).

10 202 SIGEO AKI AND KATUOMI HIRANO Ai, S., Kuboi, H. and Hirano, K. (1984). On discrete distributions of order, Ann. Inst. Statist. Math., 36, Charalambides, Ch. A. (1986). On discrete distributions of order, Ann. Inst. Statist. Math., 38, Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Left., 9, Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, New Yor. Hirano, K. (1986). Some properties of the distributions of order, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 43-53, Reidel, Dordrecht. Ling, K. D. (1988). On binomial distributions of order/c, Statist. Probab. Left., 6, Philippou, A. N. (1986). Distributions and Fibonacci polynomials of order, longest runs, and reliability of consecutive--out-of-n:f system, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), , Reidel, Dordrecht. Philippou, A, N. (1988). On multiparameter distributions of order, Ann. Inst. Statist. Math., 4O, Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties, Statist. Probab. Lett., 1, Sobel, M. and Ebneshahrashoob, M. (1992). Quota sampling for multinomial via Dirichlet, g. Statist. Plann. Inference, 33,

LONGEST SUCCESS RUNS AND FIBONACCI-TYPE POLYNOMIALS

LONGEST SUCCESS RUNS AND FIBONACCI-TYPE POLYNOMIALS ANDREAS N. PHILIPPOU & FROSSO S. MAKRI University of Patras, Patras, Greece (Submitted January 1984; Revised April 1984) 1. INTRODUCTION AND SUMMARY