˙ ˙ IK: ˙ JOURNAL OF THE TURKISH STATISTICAL ASSOCIATION ISTAT IST Vol. 9, No. 2, July 2016, pp. 67–77 issn 1300-4077 | 16 | 2 | 67 | 77
˙ ˙ IK ˙ ISTAT IST
ON THE NUMBER OF l-OVERLAPPING SUCCESS RUNS OF LENGTH k UNDER q- SEQUENCE OF BINARY TRIALS Ismail Kinaci, Co¸skun Ku¸s*, Kadir Karakaya and Yunus Akdo˘gan Department of Statistics, Faculty of Science, Selcuk University, 42250, Konya, Turkey
Abstract: Let X1 , X2 , . . . , Xn be a {0, 1}-valued Bernoulli trials with a geometrically varying success probability. The probability mass function, first and second moments of the number of l-overlapping success runs of length k in X1 , X2 , . . . , Xn are obtained. The new distribution generalizes, Type I and Type II q-binomial distributions which were recently studied in the literature. Key words : Discrete distributions; q-Distributions; Runs History : Submitted: 4 May 2016; Revised: 27 June 2016; Accepted: 8 July 2016
1. Introduction The distribution of the number of success runs has been widely studied in the literature under various assumptions on binary sequences. See, e.g. [1]-[7]. Let {Xi }i≥1 be a sequence of Bernoulli trials such that the trials of subsequences after the (i − 1)st zero until the i-th zero are independent with failure probability qi = 1 − θq i−1 , i = 1, 2, . . . , 0 < θ < 1, 0 < q ≤ 1.
(1.1)
Charalambides [8] studied discrete q− distributions under the above mentioned Bernoulli scheme. In particular, he has obtained the pmf of the number Zn of successes in X1 , X2 , . . . , Xn as n−s n sY P (Zn = s) = θ 1 − θq i−1 , s = 0, 1, . . . , n, (1.2) s q i=1 where
[n]s,q n = [s]q ! s q s
and [n]s,q = [n]q [n − 1]q × · · · × [n − s + 1]q , [s]q = 1−q , [s]q ! = [1]q [2]q × · · · × [s]q . Note that 1−q q−binomial distribution converges to the usual binomial distribution as q → 1. According to the nonoverlapping enumeration scheme, the distribution of the number of success runs of length k in n trials follows a Type I binomial distribution of order k which reduces to the well-known binomial distribution when k = 1. Recently, Yalcin and Eryilmaz [9] and Yalcin [10] introduced and studied q-geometric distribution of order k and q-binomial distribution of order k under q-sequence of binary trials. q In this study, we consider a new generalization of a q-binomial distribution of order k. Let Nn,k,l denote the number of l-overlapping success runs of length k in n Bernoulli trials under q-sequence of binary trials. For example, let n = 15 and the trials be Outcomes: 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 * Corresponding author. E-mail address:
[email protected] 67
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
68
q Then, the number of l = 2 overlapping success run of length k = 3 is N15,3,2 = 6 and the corresponding success runs are
1 2 3, 5 6 7, 6 7 8, 10 11 12, 11 12 13, 12 13 14. For l = 1 and k = 3, the corresponding success runs are 1 2 3, 5 6 7, 10 11 12, 12 13 14 q and hence N15,3,1 = 4. q The distribution of random variable Nn,k,l reduces • to the distribution of Nn,k,l (Makri and Philippou [4]) when q → 1 • to the distribution of Mn,k (Yalcin [10]) when l = k − 1 • to the distribution of Mn(k) (Ling [3]) when q → 1 and l = k − 1 • to the distribution of Nn,k (Yalcin and Eryilmaz [9]) when l = 0 • to the distribution of Nn(k) (Hirano [1] and Philippou and Makri [2]) when q → 1 and l = 0. q is The paper is organized as follows. In Section 2, probability mass function (pmf) of Nn,k,l obtained. First and second moments of introduced distribution are also derived.
q 2. Distribution and the first two moments of Nn,k,l We first prove the following lemma which will be useful in the sequel.
Lemma 1. For 0 < q ≤ 1, define Aq (r, s, t) =
X
···
X
q y2 +2y3 +···+(r−1)yr ,
B
where B = {y : y1 + y2 + · · · + yr = s, c1 + c2 + · · · + cr = t, y1 ≥ 0, . . . , yr ≥ 0} , (h cj =
yj −l k−l
i
0,
, if yj ≥ k otherwise,
(2.1)
y = (y1 , y2 , . . . , yr ) , [x] denotes the integer part of x and yi s (i = 1, 2, . . . , r) are all integers. Then Aq (r, s, t) obeys the following recurrence relation s P (r−1)j q Aq (r − 1, s − j, t − dj ) , if r > 1, s ≥ 0, t ≥ 0 j=0 s−l or if r = 1, s ≥ k, t = k−l Aq (r, s, t) = 1, (r = 1, 0 ≤ s < k, t = 0) 0, otherwise, where
(h dj =
j = 0, 1, . . . , s.
j−l k−l
0,
i
, if j ≥ k otherwise,
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
69
Proof. For r > 1, consider the values that yr can take, it can be written X X X X Aq (r, s, t) = ··· q y2 +2y3 +···+(r−2)yr−1 + q r−1 ··· q y2 +2y3 +···+(r−2)yr−1 B0 B1 X X X X 2(r−1) y2 +2y3 +···+(r−2)yr−1 +q ··· q + · · · + q s(r−1) ··· q y2 +2y3 +···+(r−2)yr−1 B2
Bs
= Aq (r − 1, s, t − d0 ) +q r−1 Aq (r − 1, s − 1, t − d1 ) + · · · + q s(r−1) Aq (r − 1, 0, t − ds ) s X = q (r−1)j Aq (r − 1, s − j, t − dj ) , j=0
where Bj = {y : y1 + y2 + · · · + yr−1 = s − j, c1 + c2 + · · · + cr−1 = t − dj , y1 ≥ 0, . . . , yr−1 ≥ 0} for j = 0, 1, . . . , s such that B = ∪sj=0 Bj . The other parts of recurrence are obvious. q Theorem 1. Let Nn,k,l denote the number of l-overlapping success runs of length k in n Bernoulli trials under q-sequence of binary trials. Then , for 0 < q ≤ 1, the probability mass function q of Nn,k,l is given by
P
q Nn,k,l
ν n (x) i Y X n−i =x = θ 1 − θq j−1 Aq (i + 1, n − i, x) , i=0
x = 0, 1, . . . ,
h
n−l k−l
i
j=1
, where ν (x) =
n, if x = 0 n − (x (k − l) + l) , otherwise,
and Aq (r, s, t) is defined as in Lemma 1. Proof. Let Sn denote the total number of zeros (failures) in n trials. Then q X q P Nn,k,l =x = P Nn,k,l = x, Sn = i . i
The joint event consists of i zeros.
q Nn,k,l = x, Sn = i can be described with the following binary sequence which . . 1}0 . . . 01| .{z . . 1}01| .{z 1| .{z . . 1}01| .{z . . 1}, y1
y2
yi
yi+1
where yj ≥ 0, j = 1, 2, . . . , i + 1, y1 + y2 + · · · + yi+1 = n − i c1 + c2 + · · · + ci+1 = x and cj s are defined as Eq. (2.1).
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
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It is clear that i can take the values at least 0 and at most ν (x) . Under the model (1.1), q P Nn,k,l =x ν(x) X X X y y y y = ··· θq 0 1 1 − θq 0 (θq) 2 (1 − θq) × · · · × θq i−1 i 1 − θq i−1 θq i i+1 i=0
i+1
P
y =n−i
j j=1 i+1
P
cj = x
j=1
y1 ≥ 0, . . . , yi+1 ≥ 0
=
ν(x) X
θ
n−i
i=0
i Y
1 − θq j−1
X
j=1
···
X
q y2 +2y3 +···+(i−1)yi +iyi+1
i+1
P
y =n−i
j j=1 i+1
P
cj = x
j=1
y1 ≥ 0, . . . , yi+1 ≥ 0
=
ν(x) X
θ
n−i
i=0
i Y
1 − θq j−1 Aq (i + 1, n − i, x) .
j=1
q As a special case, the pmf of the N12,3,2 is given for different parameters settings in Table 1. Let X1 , X2 , . . . , Xn be the Bernoulli random variables with geometrically varying success i h n−k probability. For i = 1, k − l + 1, 2 (k − l) + 1, . . . , k−l (k − l) + 1, let Ei be the event that h i i−2 ”X1 X2 · · · Xi+k−1 = 1” and for i = 2, . . . , n − k + 1 and j = 0, 1, . . . , k−l , let Ei,j be the event that ”Xi−j(k−l)−1 = 0, Xi−j(k−l) · · · Xi+k−1 = 1”. Next A1 , A2 , A3 be a partition of the index set I = {1, 2, . . . , n − k + 1} , where
A1 = {1} n , h i o A2 = k − l + 1, 2 (k − l) + 1, . . . , n−k (k − l) + 1 , k−l A3 = I − (A1 ∪ A2 ) . In the following Theorem, we obtain an expression for the expected value of the random variable q . Nn,k,l q q Theorem 2. Let Nn,k,l be a random variable as in Theorem 1. Then for n ≤ k − 1, E Nn,k,l = 0 and for n ≥ k > l ≥ 0, i−2
E
q Nn,k,l
k
=θ +
X i∈A2
θ
i+k−1
+
n−k+1 k−l ] ν i,j X [X X i=2
k+j(k−l) P Zν i,j = s 1 − θq ν i,j −s θq ν i,j −s+1 ,
j=0 s=0
where ν i,j = i − j (k − l) − 2 and P (Zy = s) is defined as Eq. (1.2) Proof. Let us define random variables Yi , 1 ≤ i ≤ n − k + 1, as follows: 1, if E1 occurs Y1 = 0, otherwise, for i ∈ A2 , Yi =
1, if Ei ∪
i−2 [ k−l S]
j=0
0, otherwise,
Ei,j occurs
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
and for i ∈ A3 , Yi =
1, if
i−2 [ k−l S]
71
Ei,j occurs
j=0
0, otherwise.
q q In this case Nn,k,l can be written by using Y s as Nn,k,l = q Nn,k,l is obtained by
Pn−k+1 i=1
Yi . Therefore, expected value of
X X q E Nn,k,l = P (Y1 = 1) + P (Yi = 1) + P (Yi = 1) , i∈A2
(2.2)
i∈A3
where P (Y1 = 1) = θk ,
(2.3)
for i ∈ A2 P (Yi = 1) = P Ei,0 ∪ Ei,1 ∪ · · · ∪ Ei,[ i−2 ] ∪ Ei k−l = P (Ei,0 ) + P (Ei,1 ) + · · · + P Ei,[ i−2 ] + P (Ei )
(2.4)
P (Ei,j ) = P Xi−j(k−l)−1 = 0, Xi−j(k−l) · · · Xi+k−1 = 1 ν i,j h i X k+j(k−l) i−2 = P Zν i,j = s 1 − θq ν i,j −s θq ν i,j −s+1 , j = 0, 1, . . . , k−l ,
(2.5)
k−l
and
s=0
P (Ei ) = P (X1 · · · Xi+k−1 = 1) = θi+k−1 .
(2.6)
Hence, substituting Eqs. (2.5)-(2.6) in Eq. (2.4), for i ∈ A2
P (Yi = 1) =
i−2 [X k−l ] ν i,j X
k+j(k−l) P Zν i,j = s 1 − θq ν i,j −s θq ν i,j −s+1 + θi+k−1 .
(2.7)
j=0 s=0
Furthermore for i ∈ A3, P (Yi = 1) can be similarly obtained as
P (Yi = 1) =
i−2 [X k−l ] ν i,j X
k+j(k−l) P Zν i,j = s 1 − θq ν i,j −s θq ν i,j −s+1 .
(2.8)
j=0 s=0
The proof is completed by using Eqs. (2.3), (2.7) and (2.8) in Eq. (2.2). q q Theorem 3. Let Nn,k,l be a random variable as in Theorem 1. Then for n ≤ k −1, E Nn,k,l 0 and n ≥ k > l ≥ 0, 2 q q E Nn,k,l = E Nn,k,l + 2 (I1 + I2 ) ,
where r−k−2
I1 =
n−k+1 k−l X [ X r=2
j=0
] ηX 1,r,j s=0
k+j(k−l) X r+k−1 P Zη1,r,j = s θk 1 − θq η1,r,j −s θq η1,r,j −s+1 + θ , r∈A2
2
=
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
72
] " νX [ r−i−k−1 i,t1 η i,r,t2 k−l X k+t1 (k−l) X I2 = P Z = s 1 − θq ν i,t1 −s1 θq ν i,t1 −s1 +1 ν i,t1 1 i=2 r=i+1 t1 =0 t2 =0 s1 =0 s2 =0 k+t2 (k−l) ×P Wηi,r,t2 = s2 1 − θq ν i,t1 +ηi,r,t2 −(s1 +s2 )+1 θq ν i,t1 +ηi,r,t2 −(s1 +s2 )+2 ) ν r,t1 X r+k−i+t (k−l) 1 +δ P Zν r,t1 = s1 1 − θq ν r,t1 −s1 θq ν r,t1 −s1 +1 i−2
n−k n−k+1 k−l ] X X [X
s1 =0
+
r−i−k−1 ] ηX i,r,t2 k−l X n−k+1 X [ X
i∈A2 r=i+1
+
X X
θ
t2 =0 r+k−1
k+t2 (k−l) P Zηi,r,t2 = s θi+k−1 1 − θq ηi,r,t2 −s θq ηi,r,t2 −s+1
s=0
,
i∈A2 r∈A2 r>i
δ=
1 (k−l) 1, if r−i+t is integer k−l 0, otherwise,
ν i,j = i − j (k − l) − 2, η i,r,j = r − i − k − j (k − l) − 1. Also P (Za = s1 ) and P (Wb = s2 ) are pmf of the q−binomial distribution (see Eq. (1.2)) with a and b trials and initial success probability θ and q θq a−s1 +1 , respectively. Note that E Nn,k,l is already obtained in Theorem 2. Proof. It is clear that q Nn,k,l
2
=
n−k+1 X
!2 Yi
=
i=1
n−k+1 X
Yi2 + 2
i=1
n−k n−k+1 X X
Yi Yr
i=1 r=i+1
and one can also write q E Nn,k,l
2
n−k n−k+1 X X q = E Nn,k,l +2 E (Yi Yr ) . i=1 r=i+1
Last term in RHS of Eq. (2.9) is obtained by n−k n−k+1 X X
E (Yi Yr ) =
i=1 r=i+1
=
n−k n−k+1 X X i=1 r=i+1 n−k+1 X
P (Yi Yr = 1)
P (Y1 Yr = 1) +
r=2
n−k n−k+1 X X
P (Yi Yr = 1) ,
i=2 r=i+1
where P (Y1 Yr = 1) ] [ r−2 k−l S Er,j ∪ Er , r ∈ A2 P E1 ∩ j=0 = [ r−2 ] k−l S P E ∩ Er,j , r ∈ A3 1 j=0 r−k−2 [ k−l 1,r,j P ] ηP k+j(k−l) P Z = s θk (1 − θq η1,r,j −s ) (θq η1,r,j −s+1 ) + θr+k−1 , r ∈ A2 η 1,r,j j=0 s=0 = [ r−k−2 1,r,j k−l P ] ηP k+j(k−l) P Z = s θk (1 − θq η1,r,j −s ) (θq η1,r,j −s+1 ) , r ∈ A3 , η 1,r,j j=0
s=0
(2.9)
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
r−2 i−2 [ k−l [ ] ] k−l S S E ∪ Ei ∩ Er,t2 ∪ Er P t1 =0 i,t1 t2 =0 i−2 r−2 [ k−l [ ] ] k−l S S P E ∪ E ∩ E i,t i r,t 1 2 t1 =0 t2 =0 P (Yi Yr = 1) = r−2 i−2 ] [ ] [ k−l k−l S S E E ∪ E P ∩ i,t r,t r 1 2 t2 =0 t1 =0 r−2 i−2 ] [ ] [ k−l k−l S S E E P ∩ i,t r,t 1 2 t1 =0 t2 =0 r−i−k−1 i−2 ] ] [ [ k−l k−l P P P (Ei,t1 ∩ Er,t2 ) + δP Er, r−i+t1 (k−l) k−l t =0 t =0 1 2 , r−i−k−1 [ ] k−l P + P (Ei ∩ Er,t2 ) + P (Ei ∩ Er ) t2 =0 [ i−2 ] [ r−i−k−1 ] k−l k−l P P P (Ei,t1 ∩ Er,t2 ) + δP Er, r−i+t1 (k−l) = k−l t1 =0 t2 =0 , [ r−i−k−1 ] k−l P + P (Ei ∩ Er,t2 ) t2 =0 i−2 r−i−k−1 [ ] [ ] k−l k−l P P , P (Ei,t1 ∩ Er,t2 ) + δP Er, r−i+t1 (k−l) t1 =0 k−l t2 =0
73
, i, r ∈ A2
,
i ∈ A2 r ∈ A3
,
i ∈ A3 r ∈ A2
, i, r ∈ A3
i ∈ A2 r ∈ A2
i ∈ A2 r ∈ A3
i ∈ A3 , r ∈ A2 ∪ A3
P (Ei ∩ Er ) = θr+k−1 , η i,r,t X2 k+t2 (k−l) P (Ei ∩ Er,t2 ) = P Zηi,r,t2 = s θi+k−1 1 − θq ηi,r,t2 −s θq ηi,r,t2 −s+1 , s=0 ν i,t1 η i,r,t2
P (Ei,t1 ∩ Er,t2 ) =
X X
k+t1 (k−l) P Zν i,t1 = s1 1 − θq ν i,t1 −s1 θq ν i,t1 −s1 +1
s1 =0 s2 =0
k+t2 (k−l) ×P Wηi,r,t2 = s2 1 − θq ν i,t1 +ηi,r,t2 −(s1 +s2 )+1 θq ν i,t1 +ηi,r,t2 −(s1 +s2 )+2 , νX r,t1 r+k−i+t1 (k−l) P Er, r−i+t1 (k−l) = P Zν r,t1 = s1 1 − θq ν r,t1 −s1 θq ν r,t1 −s1 +1 , k−l
s1 =0
P (Za = s1 ) and P (Wb = s2 ) are pmf of the q−binomial distribution with initial success probability θ and θq a−s1 +1 . q q E Nn,k,l and V ar Nn,k,l are given for different values of n, k, l, q and θ in Tables 2-3 and some related figures are also given in Figs. 2-3 for n = 10. From these tables and figures, it can be seen that:
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
74
q q • When l increases, both E Nn,k,l and V ar N n,k,l increase q q • When k increases, both E Nn,k,l and V ar Nn,k,l decrease q q • When θ or q increases, E Nn,k,l increases and V ar Nn,k,l is non-monotone.
References [1] Hirano, K. (1986). Some Properties of the Distributions of Order k. Fibonacci Numbers and Their Applications (eds. G.E. Bergum, A.N. Philippou, and A.F. Horadam), 43-53. [2] Philippou, A.N. and Makri, F.S. (1986). Success Runs and Longest Runs. Statistics & Probability Letters, 4, 211-215. [3] Ling, K.D. (1988). On Binomial Distribution of Order k. Statistics & Probability Letters, 6, 247-250. [4] Makri, F.S. and Philippou, A.N. (2005). On Binomial and Circular Binomial Distributions of order k for l -overlapping success runs of length k. Statistical Papers, 46(3), 411-432. [5] Eryilmaz, S. and Demir, S. (2007). Success runs in a sequence of exchangeable binary trials. Journal of Statistical Planning and Inference, 137, 2954–2963. [6] Demir, S. and Eryilmaz, S. (2010). Run statistics in a sequence of arbitrarily dependent binary trials. Statistical Papers, 51, 959–973. [7] Eryilmaz, S. and Yalcin, F. (2011). Distribution of run statistics in partially exchangeable processes. Metrika, 73, 293–304. [8] Charalambides, C.A. (2010). The q -Bernstein basis as a q -binomial distributions. Journal of Statistical Planning and Inference, 140, 2184–2190. [9] Yalcin, F. and Eryilmaz S. (2014). q -Geometric and q -Binomial Distributions of Order k. Journal of Computational and Applied Mathematics, 271, 31–38. [10] Yalcin, F. (2013). On a Generalization of Ling’s Binomial Distribution. ISTATISTIK, Journal of the Turkish Statistical Association, 6(3), 110–115.
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77, q Table 1. Probability mass function of N12,3,2
x q = 0.5, θ = 0.5 q = 0.5, θ = 0.8 q = 0.8, θ = 0.5 0 0.8594 0.4526 0.7668 1 0.0733 0.1169 0.1243 2 0.0343 0.0888 0.0570 3 0.0166 0.0694 0.0270 4 0.0082 0.0550 0.0130 5 0.0041 0.0438 0.0062 6 0.0020 0.0350 0.0030 7 0.0010 0.0312 0.0016 8 0.0005 0.0215 0.0004 9 0.0002 0.0172 0.0003 10 0.0002 0.0687 0.0002
q Table 2. E Nn,k,l for different values of θ, q, n, k and l n
l k (0.5, 0.2) 0.34237 0.16504 0.08212 0.03907 0.05860
(θ, q) (0.5, 0.5) 0.41692 0.18209 0.08608 0.03996 0.05968
(0.5, 1) 1.22265 0.52734 0.22265 0.08984 0.12499
(0.8, 0.5) 1.64504 1.10233 0.85677 0.54280 0.97420
0 1 2 3 4
2 3 4 5 5
(0.2, 0.5) 0.05509 0.00948 0.00177 0.00034 0.00041
10 0 1 2 3 4
2 3 4 5 5
0.05510 0.00948 0.00177 0.00034 0.00041
0.34348 0.16702 0.08310 0.04102 0.06153
0.42003 0.18471 0.08734 0.04204 0.06279
1.55566 0.69433 0.30566 0.13183 0.18749
1.78102 1.24823 0.96948 0.67952 1.21949
12 0 1 2 3 4
2 3 4 5 5
0.05510 0.00948 0.00177 0.00034 0.00041
0.34375 0.16752 0.08335 0.04151 0.06226
0.42083 0.18537 0.08765 0.04256 0.06357
1.88890 0.86107 0.38891 0.17358 0.24999
1.86823 1.34169 1.04167 0.76707 1.37657
8
q Table 3. V ar Nn,k,l for different values of θ, q, n, k and l n
l k (0.5, 0.2) 0.44446 0.23177 0.12226 0.05317 0.14111
(θ, q) (0.5, 0.5) 0.60066 0.25033 0.12588 0.05407 0.14252
(0.5, 1) 2.04772 0.61644 0.25120 0.10521 0.24268
(0.8, 0.5) 2.26448 1.42630 1.32089 0.66825 2.39907
0 1 2 3 4
2 3 4 5 5
(0.2, 0.5) 0.07342 0.01035 0.00191 0.00037 0.00061
10 0 1 2 3 4
2 3 4 5 5
0.08553 0.01090 0.00192 0.00037 0.00061
0.45503 0.24494 0.12894 0.06278 0.16908
0.68776 0.27380 0.13359 0.06381 0.17082
3.31330 1.29621 0.41536 0.16523 0.40429
3.09906 2.10510 1.88385 1.17570 4.25853
12 0 1 2 3 4
2 3 4 5 5
0.09760 0.01145 0.00195 0.00037 0.00061
0.45946 0.24924 0.13111 0.06616 0.17900
0.76340 0.28704 0.13700 0.06731 0.18093
4.42100 2.02928 0.78160 0.25575 0.59765
3.84258 2.71397 2.37884 1.65544 6.00807
8
75
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
76
k=3
k=4
k=5
l=3
l=2
l=4
1
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0
0 2 4 6 8 10 q=0.2
1
0
0 0 2 4 6 8 10 0 2 4 6 8 10 (n=12, l=2, q=0.5, θ=0.5) q=0.5 q=0.8 1 1
0
0 2 4 6 8 10 θ=0.5
1
0 0 0 2 4 6 8 10 0 2 4 6 8 10 (n=12,k=5,q=0.5,θ=0.5) θ=0.9 θ=0.8 1 1
0.8
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0
0 2 4 6 8 10
0
0 0 2 4 6 8 10 0 2 4 6 8 10 (n=12, k=3, l=2, θ=0.5)
0
0 2 4 6 8 10
0 0 2 4 6 8 1012 0 2 4 6 8 1012 (n=12, k=3, l=2, q=0.5) 0
q Figure 1. Probability mass function of N12,k,l for different settings
q=0.2, k=4 8 6 4
4
0
0.2
0.4
0.6
θ q=0.5, k=4
0.8
1
4
0
4 2
4 2
0
0 0.2
0.4
0.6
θ q=0.9, k=4
0.8
1
0
0.2
8
8
l=0 l=1 l=2 l=3
6 4 2 0.2
0.4
0.6
θ q=0.5, k=6
0.8
1
0.4
4
0.6
0.8
1
0
0.2
0.4
0.6 0.8 θ q=0.5, k=8
1
l=0,1,2,3,4,5 l=6 l=7
6 4 2 0.6
θ q=0.9, k=6
0.8
1
0
0
0.2
8
0.4
0.6 θ q=0.9, k=8
0.8
1
0.8
1
l=0,1,2,3,4,5 l=6 l=7
6 4
2
θ
0
8
l=0,1,2 l=3 l=4 l=5
6
0 0
0.4
l=0,1,2 l=3 l=4 l=5
6
2 0
0.2
8
l=0 l=1 l=2 l=3
6
0
l=0,1,2,3,4,5 l=6 l=7
6
2
8
0
8
l=0,1,2 l=3 l=4 l=5
6
2 0
q=0.2, k=8
q=0.2, k=6 8
l=0 l=1 l=2 l=3
2 0
0.2
0.4
θ
0.6
0.8
1
0
0
0.2
q Figure 2. E N10,k,l for different values of θ, q, k and l
0.4
θ
0.6
Kinaci et al.: On the number of l-overlapping success runs of length k under q-sequence of binary trials ˙ ˙ IK: ˙ ˙ c 2016 Istatistik ISTAT IST Journal of the Turkish Statistical Association 9(2), pp. 67–77,
q=0.2, k=4 10
l=0 l=1 l=2 l=3
8 6 4
6 4 2
0
0
0.2
10
0.4
0.6
θ q=0.5, k=4
0.8
1
6 4
0
10
0.4
0.6 θ q=0.9, k=4
0.8
1
6 4
0
0.4
θ
0.6
0.8
1
0.4
0.6
θ q=0.5, k=6
0.8
1
0
0
0.2
10
0.4
0.6 θ q=0.5, k=8
0.8
1
0.8
1
0.8
1
l=0,1,2,3,4,5 l=6 l=7
8 6 4
0.4
0.6 θ q=0.9, k=6
0.8
1
0
0
0.2
10
l=0,1,2 l=3 l=4 l=5
4
0
0.2
0.2
6 2
0
0
8
2
4
2
10
l=0 l=1 l=2 l=3
8
6
l=0,1,2 l=3 l=4 l=5
4
0
0.2
0.2
6 2
0
0
8
2
l=0,1,2,3,4,5 l=6 l=7
8
2
10
l=0 l=1 l=2 l=3
8
10
l=0,1,2 l=3 l=4 l=5
8
2 0
q=0.2, k=8
q=0.2, k=6 10
77
0.4
0.6 θ q=0.9, k=8 l=0,1,2,3,4,5 l=6 l=7
8 6 4 2
0
0.2
0.4
θ
0.6
0.8
1
0
0
0.2
q Figure 3. V ar N10,k,l for different values of θ, q, k and l
0.4
θ
0.6
