International Journal of Statistical Distributions and Applications 2016; 2(1): 8-13 http://www.sciencepublishinggroup.com/j/ijsda doi: 10.11648/j.ijsd.20160201.12 ISSN: 2472-3487 (Print); ISSN: 2472-3509 (Online) Extended Intervened Geometric Distribution C. Satheesh Kumar, S. Sreejakumari Department of Statistics, University of Kerala, Trivandrum, India Email address: drcsatheeshkumar@gmail.com (C. S. Kumar), kumari.sreeja@gmail.com (S. Sreejakumari) To cite this article: C. Satheesh Kumar, S. Sreejakumari. Extended Intervened Geometric Distribution. International Journal of Statistical Distributions and Applications. Vol. 2, No. 1, 2016, pp. 8-13. doi: 10.11648/j.ijsd.20160201.12 Received: March 3, 2016; Accepted: March 29, 2016; Published: April 25, 2016 Abstract: Here we develop an extended version of the modified intervened geometric distribution of Kumar and Sreeja (The Aligarh Journal of Statistics, 2014) and investigate some of its important statistical properties. Parameters of the distribution are estimated by various methods of estimation such as the method of factorial moments, the method of mixed moments and the method of maximum likelihood. The distribution has been fitted to a real life data set for illustrating its practical relevance. Keywords: Factorial Moments, Intervened Geometric Distribution, Method of Factorial Moments, Method of Mixed Moments, Method of Maximum Likelihood, Probability Generating Function, Probability Mass Function 1. Introduction Intervened type distributions have found many applications in several areas such as epidemiological studies, life testing problems etc. In epidemiological studies health agencies takes various preventive actions. The information concerning the effect of such actions taken by the agencies can statistically analyzed by intervened type distributions. In life testing problem the failed items during the observational period are either replaced or repaired. This kind of actions changes the reliability of the system as only some of its components have longer life. The impact of such actions can be studied by intervened type distributions. The intervened type distributions such as intervened Poisson distribution (IPD), intervened geometric distribution (IGD) and modified intervened geometric distribution (MIGD) has been studied by several authors. For example see Shanmugan [1, 2], Huang and Fung [3], Scollink [4], Dhanavanthan [5, 6], Kumar and Shibu [7-15], Bartolucci et al [16], Kumar and Sreeja [17] etc. Through this paper we consider a new class of intervened geometric distribution suitable for multiple intervention cases and named it as the extended intervened geometric distribution (EIGD), which contains the MIGD as its special case. The paper is organized as follows. In Section 2, we present a model leading to EIGD and obtain expression for its probability mass function, mean and variance. We also obtain a recurrence relation useful for the computation of probabilities of the EIGD. In Section 3, we consider the estimation of parameters of the EIGD by the method of maximum likelihood and the distribution has been fitted to a real life data set for highlighting the usefulness of the model. We need the following series representation in the sequel i i 0r 0 i 0r 0 A(i,r) A(i r,r) ∞ ∞ ∞ = = = = = - ∑∑ ∑∑ (1) and i m i 0r 0 i 0r 0 B(i,r) B(i mr,r),     ∞ ∞ ∞   = = = = = - ∑∑ ∑∑ (2) where [a] represents the integer part of “a”, for any a > 0 2. Extended Intervened Geometric Distribution Consider a discrete random variable X having intervened geometric distribution with the following probability mass function (pmf), in which x = 1, 2, 3,. . . , θ∈(0, 1), 1 1 0 ≠ρ > such that 1 1 ρθ≤ .