ass="t m0 x0 he y6d ff1 fs2 fc0 sc0 ls1 wsf">1s  i.e.  , 
  Such a sequence, called a finite Bernoulli sequence,
is of particular importance in studies of applied probability
because of its simplicity, and also since it may be considered as
a special case of a sequence with dependent elements; e.g. a
Markovian or an exchangeable one.
For a Bernoulli sequence of length , let   
denote the probability mass function (PMF) of the RV 
; i.e.
 
 ,   
,  . (2)
Then, the expected value of 
,
  
   

 (3)
is given by (see Makri et al. [11])
 

   
  
 
 
   
In the sequel, we consider a symmetric () finite
Bernoulli sequence (i.e. a finite binary string) of length for
which we obtain our main results. Since the cardinality of a
proper sample space is (i.e. there are binary strings that
are equally likely to occur) the classical definition of
probability implies that
  
 

. (5)
The numbers 
 ,  
, for  admit the
following interpretation: (i) 
 ,   is the number of
all binary strings of length with exactly  

 , 1-runs of length [exactly (),
at least ()] , among all the possible binary strings of
length , . (ii) 
 is the number of all binary
strings of length with exactly  n, 1s
contained in all 1-runs of length at least , among all the
possible binary strings of length , .
Simple explicit expressions of 
 (not repeated here) are
given in Makri and Psillakis [19]. The authors provided an
explicit formula of 
 , for  in terms of binomial
coefficients. Their method is based on the solution of a
combinatorial problem; specifically, the allocation of balls into
cells under certain constrains (see Lemma 2.2 of [11]). An
explicit expression of 
 , in terms of binomial coefficients
too, is given by Sinha and Sinha [1] who used a generating
function approach. The latter expression contains an additional
sum; therefore it may be evaluated slower computationally than
that provided in [19].
The numbers 
 allow us to establish (we do not actually
need their specific expressions according to the new proposed
approach) respective numbers referring to all possible
binary strings of length . They are defined as




, (6)
i.e. 
 is the total number of occurrences of all 1-runs of
length [exactly (), at least ()] , and the total
number of 1s in all 1-runs of length at least [], in all
possible binary strings of length , .
Since 
  

, hence
by (5) and (6), 
 
. Therefore (4) implies


   
  
  
 
  
(7)
Readily, by symmetry, 
 provides the respective
numbers associated with 0-runs and 0s in 0-runs.
By (7) it is noted that for a fixed , 
,  
decreases exponentially as increases, and for a fixed 
, as , it holds

 

 
 

 


 
(8)
Furthermore, 
 

 , since   ,
.
Table I presents the three numbers 


 in binary
strings of length  bits,  , and for . The
entries of the table confirm the previously noted behavior of
the depicted numbers.
Sinha [31] was the first who addressed the usefulness of the
number 
 and also provided its formula. Then, Sinha and
Sinha [1] obtained an explicit expression of 
 whereas
Copyright © 2012 SciRes.
45
Makri and Psillakis [19] derived the same formula for it and
they also established an explicit expression of 
. In both
papers ([1] and [19]) the approach was relied on the definition
of 
 via 
 , . Recently, Sinha and Sinha [2]
reestablished 
 solving explicitly a recursive generation
scheme for it.
The proposed, in the present note, approach is a new one
and treats under the same frame all the numbers 
 in a
simple, unified and systematic way. Accordingly, by (7) we
effortless get a recursive scheme for 
,  . It gives a
way to generate 
 from 
 and it offers further insight in
understanding the interdependencies among the studied
numbers. Specifically, it holds


    
 
 
 
  
(9)
with 
   . We note that
for the particular case  we capture by the relevant entries
of (9) and (7), Theorems 2 and 3 of [2], respectively.
TABLE I. NUMBERS OF OCCURRENCES OF 1-RUNS, 

 AND
NUMBER OF 1S, 
, IN BINARY STRINGS OF LENGTH
4
1
12
20
32
2
5
8
20
3
2
3
10
4
1
1
4
16
1
147456
278528
524288
2
69632
131072
376832
3
32768
61440
237568
4
15360
28672
139264
5
7168
13312
77824
6
3328
6144
41984
7
1536
2816
22016
8
704
1280
11264
9
320
576
5632
10
144
256
2752
11
64
112
1312
12
28
48
608
13
12
20
272
14
5
8
116
15
2
3
46
16
1
1
16
3. Conlusions
In this note we stated three run statistics which are
important in many areas of applied probability. We defined
them on a binary (0-1) sequence, and we then provided
explicitly their mean values for a Bernoulli sequence. After that,
we considered binary strings (symmetric Bernoulli sequences)
and we showed how the analytic expressions of the means of
these RVs provide eventually the respective explicit
expressions of three numbers studied recently by different
methods. Finally, as a byproduct of our approach, we proposed
a unified recursive scheme which clarifies further the
interdependencies among these numbers. The examined
numbers are potential useful in many engineering applications
like the ones mentioned briefly in the Introduction. Early
results are encouraging in this direction.
REFERENCES
[1] K. Sinha and B. P. Sinha, “On the distribution of runs of
ones in binary strings,” Comput. Math. Appl., vol. 58, pp.
1816-1829, 2009.
[2] K. Sinha and B. P. Sinha, “Energy-efficient
communication: understanding the distribution of runs in
binary strings,” 1st International Conference on Recent
Advances in Information Technology (RAIT-2012), pp.
177-181, 2012.
[3] G. Benson, “Tandem repeats finder: a program to analyze
DNA sequences,” Nucleic Acids Res., vol. 27, pp. 573-
580, 1999.
[4] W.Y. W. Lou, “The exact distribution of the “-tuple
statistic for sequence homoloy,” Statist. Probab. Lett., vol.
61, pp. 51-59, 2003.
[5] G. Nuel, L. Regad, J. Martin and A.C. Camproux, “Exact
distribution of a pattern in a set of random sequences
generated by a Markov source: applications to biological
data,” Alg. Mol. Biol., vol. 5, pp. 1-18, 2010.
[6] N. Balakrishnan and M. V. Koutras, Runs and Scans with
Applications, New York: Wiley, 2002.
[7] J. C. Fu and W. Y. W. Lou, Distribution Theory of Runs
and Patterns and its Applications: A Finite Markov
Imbedding Approach, New Jersey: World Scientific, 2003.
[8] M. V. Koutras, “Applications of Markov chains to the
distribution theory of runs and patterns,” in Handbook of
Statistics, vol. 21, D. N. Shanbhag, C. R. Rao, Eds. North
Holland: Elsevier, 2003, pp. 431-472.
[9] S. Eryilmaz, “Success runs in a sequence of exchangeable
binary trials,” J. Statist. Plann. Inference, vol. 137, pp.
2954-2963, 2007.
[10] F. S. Makri, A.N. Philippou and Z. M. Psillakis, “Polya,
inverse Polya, and circular Polya distributions of order
for -overlapping success runs,” Commun. Statist. Theory
Methods, vol. 36, pp. 657-668, 2007.
[11] F. S. Makri, A. N. Philippou and Z. M. Psillakis, “Success
run statistics defined on an urn model,” Adv. Appl.
Probab., vol. 39, pp. 991-1019, 2007.
[12] S. Eryilmaz, “Run statistics defined on the multicolor urn
model,” J. Appl. Probab., vol. 45, pp. 1007-1023, 2008.
[13] S. Demir and S. Eryilmaz, “Run statistics in a sequence of
arbitrarily dependent binary trials,” Stat. Papers, vol. 51,
pp. 959-973, 2010.
[14] K. Inoue and S. Aki, “On the conditional and
unconditional distributions of the number of success runs
on a circle with applications,” Statist. Probab. Lett., vol.
80, pp. 874-885, 2010.
46
Copyright © 2012 SciRes.
[15] F. S. Makri, “On occurences of  strings in linearly
and circularly ordered binary sequences,” J. Appl. Probab.,
vol. 47, pp. 157-178, 2010.
[16] F. S. Makri, “Minimum and maximum distances between
failures in binary sequences,” Statist. Probab. Lett., vol.
81, pp. 402-410, 2011.
[17] F. S. Makri and Z. M. Psillakis, “On runs of length
exceeding a threshold: normal approximation,” Stat.
Papers, vol. 52, pp. 531-551, 2011.
[18] F. S. Makri and Z. M. Psillakis, “On success runs of
length exceeded a threshold,” Methodol. Comput. Appl.
Probab., vol. 13, pp. 269-305, 2011.
[19] F. S. Makri and Z. M. Psillakis, “On success runs of a
fixed length in Bernoull sequences: exact and aymptotic
results,” Comput. Math. Appl., vol. 61, pp. 761-772, 2011.
[20] S. D. Dafnis, A. N. Philippou and D. L. Anztoulakos,
“Distributions of patterns of two successes separeted by a
string of  failures,” Stat. Papers, vol. 53, pp. 323-
344, 2012.
[21] M. V. Koutras and F. S. Milienos, “Exact and asymptotic
results for pattern waiting times,” J. Statist. Plann.
Inference, vol. 142, pp. 1464-1479, 2012.
[22] F. S. Makri and Z. M. Psillakis, “Counting certain binary
strings,” J. Statist. Plann. Inference, vol. 142, pp. 908-924,
2012.
[23] A. M. Mood, “The distribution theory of runs,” Ann.
Math. Stat., vol. 11, pp. 367-392, 1940.
[24] J. C. Fu and M. V. Koutras, “Distribution theory of runs:
a Markov chain approach,” J. Amer. Statist. Assoc., vol.
89, pp. 1050-1058, 1994.
[25] D. L. Antzoulakos, “On waiting time problems associated
with runs in Markov dependent trials,” Ann. Inst. Statist.
Math., vol. 51, pp. 323-330, 1999.
[26] Q. Han and S. Aki, “Joint distributions of runs in a
sequence of multi-trials,” Ann. Inst. Statist. Math., vol. 51,
pp. 419-447, 1999.
[27] J. C. Fu, W. Y. W. Lou, Z. Bai and G. Li, “The exact and
limiting distributions for the number of successes in
success runs within a sequence of Markov-dependent
two-state trials,” Ann. Inst. Statist. Math., vol. 54, pp.
719-730, 2002.
[28] K. Sen, M. L. Agarwal and S. Chakraborty, “Lenths of
runs and waiting time distributions by using Polya-
Eggenberger sampling scheme,” Studia Sci. Math.
Hungar., vol. 2, pp. 309-332, 2002.
[29] D. L. Antzoulakos, S. Bersimis and M. V. Koutras, “On
the distribution of the total number of run lengths,” Ann.
Inst. Statist. Math., vol. 55, pp. 865-884, 2003.
[30] D. E. Martin, “Distribution of the number of successes in
success runs of length at least in higher-order
Markovian sequences,” Methodol. Comput. Appl. Probab.,
vol. 7, pp. 543-554, 2005.
[31] K. Sinha, Location and communication issues in mobile
networks. Ph. D. Dissertaion, Department of Computer
Science and Engineering, Jadavpur University, Calcutta,
India , 2007.
Copyright © 2012 SciRes.
47