ORDER STATISTICS FROM INDEPENDENT NON-IDENTICALLY DISTRIBUTED DISCRETE RAYLEIGH DISTRIBUTION

This study deals with the subject of Order Statisti cs (OS) and its moments for independent non-identic ally distributed discrete random variables rv’s drawn fr om one parameter discrete Rayleigh distribution or dRayleigh ( θ). Distributions of single order statistics and the ir moments for dRayleigh ( θ) are studied. The k moments of single os and the mean of the largest a nd smallest os are obtained for n = 3.


INTRODUCTION
The subject of order statistics for Independent Identically Distributed (IID) discrete random variates is referred to the work published by (Raider, 1951;Abdel-Aty, 1954;Siotani, 1956;Kabe, 1969;Arnold et al., 1992;Nagaraga, 1992). Not much attention has been paid to the case when assuming that the random variables are discrete and Independent Non-Identically Distributed (INID). Basic distribution theory of order statistics for this case was first discussed by (Gungor et al., 2009). The appearance of new discrete distributions in the literature such as: discrete gamma distribution (Chakraborty, 2012;Alhazzani, 2012), discrete burr and pareto distribution (Krishna and Pundir, 2009), discrete Weibull distribution (Nakagawa and Osaki, 1975;Stein and Dattero, 1984;Khan et al., 1989), discrete normal distribution (Roy, 2003) and discrete Rayleigh distribution (Roy, 2004) opens the door for lots of scientific papers in order statistics for both iid and inid cases.
Let X 1 ,X 2 ,…,X n be independent non-identically distributed (inid) random variables having adiscrete cumulative distribution function cdf F i (x) and probability mass function pmf f i (x) respectively. Let X 1:n ≤X 2:n ≤…≤X n:n denotes the order statistics obtained by arranging the n of Xi's in increasing order of magnitude and F r:n (x) (r = 1,2,..,n) be the cdf of the r th order statistics X r:n .
In this study we will study os from discrete Rayleigh distribution dRayleigh (θ) when the X i 's are inid random variables.
The pmf of r th os; X r:n from iniddRayleigh (θ) distribution is given by Equation 5: Using permanent notation the above pmf can be written as (Vaughan and Venables, 1972) when, k = m = 0 i.e., the chance of ties is zero and all the observations are distinct, an easier form of the pmf of X r:n is now obtained Equation 7 and 8:

Distribution of the Smallest and Largest os Following inid dRayleigh (θ θ θ θ) rv's
The pmf of X 1:n and X n:n from inid dRayleigh (θ) are given by Equation 9 and 10:

Moments of os from Inid dRayleigh (θ θ θ θ) rv's
Calculations of the mean of the smallest and largest os from dRayleigh (θ) are given in Table 1 and 2 for several values of 1, 2, 3.

CONCLUSION
Discrete Rayleigh distribution under contaminations (INID) is introduced. Exact forms of the pmf and cdf of order statistics and single moment are derived. Further studies are suggested for other discrete distributions.