İSTATİSTİK: JOURNAL OF THE TURKISH STATISTICAL ASSOCIATION
Vol. 9, No. 2, July 2016, pp. 67–77
issn 1300-4077 | 16 | 2 | 67 | 77
İSTATİSTİK
ON THE NUMBER OF l-OVERLAPPING SUCCESS
RUNS OF LENGTH k UNDER q- SEQUENCE OF
BINARY TRIALS
Ismail Kinaci, Coşkun Kuş*, Kadir Karakaya and Yunus Akdoğan
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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: Journal of the Turkish Statistical Association 9(2), pp. 67–77, 70
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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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
c 2016 İstatistik
İSTATİSTİK: 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