Survey of Discrete Distributions of Order k and Related Distributions
نویسنده
چکیده
The distribution of the number of trials until the first occurrence of consecutive k successes in Bernoulli trials with success probability p is called a geometric distribution of order k. This definition is due to Philippou, Georghiou and Philippou (1983). Though the distribution seems to have been interested since De Moivre (1667-1754)’s works (Todhunter (1865), Feller (1968), Johnson, Kotz and Kemp (1992, pp. 426-432) and Balakrishnan and Koutras (2002)), Philippou et al. (1983)’s contribution in this field seems to be very important. Since Philippou et al. (1983) called the distribution a geometric distribution of order k and defined a Poisson distribution of order k and a negative binomial distribution of order k, attention has been paid to interrelationships among the so called discrete distributions of order k and exact distribution theory has been developed extensively by many researchers. From early 80’s, reliability theory of the consecutive-k-out-of-n:F systems began to be studied (Kontoleon (1980), Chiang and Niu (1981) and Derman, Lieberman and Ross (1982)), and a large number of studies have been made as relationships between the reliability of the system and discrete distributions of order k have been clear (e.g. Lambiris and Papastavridis (1985), Fu (1985, 1986a, 1986b), Fu and Beihua (1987), Aki (1985), Hirano (1986),
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