On k-Gamma and k-Beta Distributions and Moment Generating Functions

Gauhar Rahman, Shahid Mubeen, Abdur Rehman, and Mammona Naz Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan Correspondence should be addressed to Gauhar Rahman; gauhar55uom@gmail.com Received 10 February 2014; Revised 29 June 2014; Accepted 4 July 2014; Published 15 July 2014 Academic Editor: Chin-Shang Li Copyright © 2014 Gauhar Rahman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The main objective of the present paper is to define k-gamma and k-beta distributions and moments generating function for the said distributions in terms of a new parameter k > 0. Also, the authors prove some properties of these newly defined distributions.

The function ()  defined in relation (1) is also known as Pochhmmer symbol.
1.2.Gamma Function.Let  ∈ C; the Euler gamma function is defined by and the integral form of gamma function is given by From the relation (3), using integration by parts, we can easily show that Γ ( + 1) = Γ () .
1.4.Pochhammer -Symbol.For  > 0, the Pochhammer symbol is denoted and defined by 2 Journal of Probability and Statistics 1.5.-Gamma Function.For  > 0 and  ∈ C, the -gamma function is defined as and the integral representation of -gamma function is 1.6.-Beta Function.For Re(), Re() > 0, the -beta function of two variables is defined by and, in terms of -gamma function, -beta function is defined as Also, the researchers [6][7][8][9][10] have worked on the generalized -gamma and -beta functions and discussed the following properties: Using the above relations, we see that, for ,  > 0 and  > 0, the following properties of -beta function are satisfied by authors (see [6,7,11]): Note that when  → 1,   (, ) → (, ).
For more details about the theory of -special functions like -gamma function, -beta function, -hypergeometric functions, solutions of -hypergeometric differential equations, contegious functions relations, inequalities with applications and integral representations with applications involving -gamma and -beta functions and so forth.(See [12][13][14][15][16][17] In statistics, there are three types of moments which are (i) moments about any point  = , (ii) moments about  = 0, and (iii) moments about mean position of the given data.Also, expected value of the variate is defined as the first moment of the probability distribution about  = 0 and the th moment about mean of the probability distribution is defined as (  − )  where  is the mean of the distribution.
Also, () shows the expected value of the variate  and is defined as the first moment of the probability distribution about  = 0; that is, 1.8.Gamma Distribution.A continuous random variable  is said to have a gamma distribution with parameter  > 0, if its probability distribution function is defined by and its distribution function () is defined by which is also called the incomplete gamma function.

Moment Generating Function of Gamma Distribution.
The moment generating function of  is defined by 1.10.Beta Distribution of the First Kind.A continuous random variable  is said to have a beta distribution with two parameters  and , if its probability distribution function is defined by This distribution is known as a beta distribution of the first kind and a beta variable of the first kind is referred to as  1 (, ).Its distribution function () is given by and its probability distribution function is given by

Main Results: 𝑘-Gamma and 𝑘-Beta Distributions
In this section, we define gamma and beta distributions in terms of a new parameter  > 0 and discuss some properties of these distributions in terms of .
Definition 1.Let  be a continuous random variable; then it is said to have a -gamma distribution with parameters  > 0 and  > 0, if its probability density function is defined by and its distribution function   () is defined by Proposition 2. The newly defined Γ  () distribution satisfies the following properties.
(i) The -gamma distribution is the probability distribution that is area under the curve is unity.(ii) The mean of -gamma distribution is equal to a parameter .(iii) The variance of -gamma distribution is equal to the product of two parameters .

Proof of (i).
Using the definition of -gamma distribution along with the relation (10), we have (30) Proof of (ii).As mean of a distribution is the expected value of the variate, so the mean of the -gamma distribution is given by Using the definition of -gamma function and the relation ( 13), we have Proof of (iii).As variance of a distribution is equal to ( 2 ) − (()) 2 , so the variance of -gamma distribution is calculated as Now, we have to find   ( 2 ), which is given by Thus we obtain the variance of -gamma distribution as where  2  is the notation of variance present in the literature.

𝑘-Beta Distribution of First Kind.
Let  be a continuous random variable; then it is said to have a -beta distribution of the first kind with two parameters  and , if its probability distribution function is defined by In the above distribution, the beta variable of the first kind is referred to as  1, (, ) and its distribution function   () is given by Proposition 3. The -beta distribution  1, (, ) satisfies the following basic properties.
(i) -beta distribution is the probability distribution that is the area of  1, (, ) under a curve   () is unity.(ii) The mean of this distribution is /( + ).
Proof of (i).By using the above definition of -beta distribution, we have By the relation (11), we get Proof of (ii).The mean of the distribution,   1, , is given by Using the relations ( 12), (13), and ( 16), we have Proof of (iii).The variance of  1, (, ) is given by Thus substituting the values of   ( 2 ) and   () in (42) along with some algebraic calculations we have the desired result.

𝑘-Beta Distribution of the Second Kind.
A continuous random variable  is said to have a -beta distribution of the second kind with parameters  and , if its probability distribution function is defined by Note.The -beta distribution of the second kind is denoted by  2, (, ).
Theorem 4. The -beta function of the second kind represents a probability distribution function that is Proof.We observe that Let 1 +  = 1/, so that  = −/ 2 ; thus by using the relation (11), the above equation gives (47)

Higher Moment in terms of 𝑘.
The th moment in terms of  is given by Theorem 5.The moments of the higher order of -beta distribution of the second kind are given as

Conclusion
In this paper the authors conclude that we have the following.
(i) If  tends to 1, then -gamma distribution and beta distribution tend to classical gamma and beta distribution.
(ii) The authors also conclude that the area of -gamma distribution and -beta distribution for each positive value of  is one and their mean is equal to a parameter  and /( + ), respectively.The variance of gamma distribution for each positive value of  is equal to  times of the parameter .In this case if  = 1, then it will be equal to variance of gamma distribution.The variance of -beta distribution for each positive value of  is also defined.
(iii) In this paper the authors introduced moments generating function and higher moments in terms of a new parameter  > 0.
.) 1.7.Probability Distribution and Expected Values.In a random experiment with  outcomes, suppose a variable  assumes the values  1 ,  2 ,  3 , . . .,   with corresponding probabilities  1 ,  2 ,  3 , . . .,   ; then this collection is called probability distribution and Σ  = 1 (in case of discrete distributions).Also, if () is a continuous probability distribution function defined on an interval [, ], then ∫ ) 1.11.Beta Distribution of the Second Kind.A continuous random variable  is said to have a beta distribution of the second kind with parameters  and , if its probability distribution function is defined by