Shortest and longest length of success runs in binary sequences

Success run statistics defined on an urn model

“…We end this section by noting that the present paper generalizes, unifies, and/or provides alternative formulae for the results of Mood (1940), Philippou et al (1983), Panaretos and Xekalaki (1986), Ling (1988), Godbole (1990), Goldstein (1990), Hirano and Aki (1993), Koutras and Alexandrou (1997), Tripsiannis and Philippou (1997), Sen et al (2002), (2003), Antzoulakos et al (2003aAntzoulakos et al ( ), (2003b, and Makri et al (2007aMakri et al ( ), (2007b.…”

“…For an alternative derivation of the distribution of L c n , see Makri et al (2007b). By conditioning on S n,1 we obtain 11) which is equivalent to Equation (2.80) of Balakrishnan and Koutras (2002).…”

“…For alternative formulae for the distributions of L n and W k , see Makri et al (2007aMakri et al ( ), (2007b.…”

Recursion Formula for the Probability Distribution of Sum of (vu+ 1)-sided die and some of its Properties

“…The number ( , ) has been used extensively in probability studies (De Moivre (1756), Feller (1968, Balakrishnan et al (1993), Balasubramanian (1995), Makri and Philippou (2005, 2007a, 2007b and related areas like reliability and inferential statistics Gabai (1970), Bollinger and Burchard (1990), Ailing (1993). For more properties on ( , ); generalized Pascal triangles or Pascal triangles of order , we refer to Fraund (1956), Gabai (1970), Bondarenko (1993), Ollerton and Shannon (1998, 2004, Dafnis (2007) and the references therein.…”

Mean success run length

“…The topic is still very attractive because of the wide range of its applications. Some recent contributions to the topic are in the works of Antzoulakos and Boutsikas (2007), Eryilmaz (2008), Eryilmaz and Demir (2007), Fu and Lou (2007), Inoue and Aki (2005), Makri, Philippou, and Psillakis (2007a) and Makri, Philippou, and Psillakis (2007b).…”