J. Stat. Appl. Pro. 4, No. 1, 119-126 (2015) 119 Journal of Statistics Applications & Probability An International Journal http://dx.doi.org/10.12785/jsap/040112 Generalization of Chi-square Distribution Gauhar Rahman 1 , Shahid Mubeen 2,∗ and Abdur Rehman 2 1 International Islamic University Islamabad, Pakistan 2 University of Sargodha, Sargodha, Pakistan Received: 10 Aug. 2014, Revised: 15 Feb. 2015, Accepted: 18 Feb. 2015 Published online: 1 Mar. 2015 Abstract: In this paper, we define a generalized chi-square distribution by using a new parameter k > 0. we give some properties of the said distribution including the moment generating function and characteristic function in terms of k. Also, we establish a relationship in central moments involving the parameter k > 0. If k = 1, we have all the results of classical χ 2 distribution. Keywords: k-gamma functions, chi-square distribution, moments 1 Introduction and basic definitions The chi-square distribution was first introduced in 1875 by F.R. Helmert, a German physicist. Later in 1900, Karl Pearson proved that as n approaches infinity, a discrete multinomial distribution may be transformed and made to approach a chi-square distribution. This approximation has broad applications such as a test of goodness of fit, as a test of independence and as a test of homogeneity. The chi-square distribution contains only one parameter, called the number of degrees of freedom, where the term degree of freedom represent the number of independent random variables that express the chi-square. If the random variables entering a chi-square are subjected to linear restrictions, then the number of degrees of freedom is reduced by the number of restrictions involved. we generalize the chi-square distribution in the form of a new parameter k where k > 0. Here, we give some definitions which provide a base for our main results. The definitions (1.1 − 1.2) are given in [1] while (1.3 − 1.4) are introduced in [2]. Also, we have taken some statistics related definitions (1.5 − 1.11) from [3-6]. 1.1 Pochhmmer’s Symbol. The factorial function is denoted and defined by (a) n =  a(a + 1)(a + 2) ··· (a + n − 1); for n ≥ 1, a = 0 1 if n = 0. (1.1) The function (a) n defined in relation (1.1) is also known as Pochhmmer’s symbol. 1.2 Gamma Function. Let z ∈ C, the Euler gamma function is defined by Γ (z)= lim n→∞ n!n z−1 (z) n (1.2) ∗ Corresponding author e-mail: smjhanda@gmail.com c  2015 NSP Natural Sciences Publishing Cor.