Journal of Probability and Statistics

Volume 2014, Article ID 982013, 6 pages

http://dx.doi.org/10.1155/2014/982013

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

Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan

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.

#### Abstract

The main objective of the present paper is to define -gamma and -beta distributions and moments generating function for the said distributions in terms of a new parameter . Also, the authors prove some properties of these newly defined distributions.

#### 1. Basic Definitions

In this section we give some definitions which provide a base for our main results. The definitions (1.1–1.3) are given in [1] while (1.4–1.6) are introduced in [2]. Also, we have taken some statistics related definitions (1.7–1.11) from [3–5].

##### 1.1. Pochhmmer Symbol

The factorial function is denoted and defined by The function defined in relation (1) is also known as Pochhmmer symbol.

##### 1.2. Gamma Function

Let ; 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 The relation between Pochhammer symbol and gamma function is given by

##### 1.3. Beta Function

The beta function of two variables is defined as and, in terms of gamma function, it is written as

##### 1.4. Pochhammer -Symbol

For , the Pochhammer -symbol is denoted and defined by

##### 1.5. -Gamma Function

For and , the -gamma function is defined as and the integral representation of -gamma function is

##### 1.6. -Beta Function

For , the -beta function of two variables is defined by and, in terms of -gamma function, -beta function is defined as

Also, the researchers [6–10] have worked on the generalized -gamma and -beta functions and discussed the following properties: Using the above relations, we see that, for and , the following properties of -beta function are satisfied by authors (see [6, 7, 11]): Note that when , .

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 [12–17].)

##### 1.7. Probability Distribution and Expected Values

In a random experiment with outcomes, suppose a variable assumes the values with corresponding probabilities ; then this collection is called probability distribution and (in case of discrete distributions). Also, if is a continuous probability distribution function defined on an interval , then .

In statistics, there are three types of moments which are (i) moments about any point , (ii) moments about , 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 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 ; that is,

##### 1.8. Gamma Distribution

A continuous random variable is said to have a gamma distribution with parameter , if its probability distribution function is defined by and its distribution function is defined by which is also called the incomplete gamma function.

##### 1.9. 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 . Its distribution function is given by

##### 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 and its probability distribution function is given by

#### 2. Main Results: -Gamma and -Beta Distributions

In this section, we define gamma and beta distributions in terms of a new parameter 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 and , if its probability density function is defined by
and its distribution function is defined by

Proposition 2. *The newly defined 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

*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 , so the variance of -gamma distribution is calculated as
Now, we have to find , which is given by
Thus we obtain the variance of -gamma distribution as
where is the notation of variance present in the literature.

*2.1. -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 and its distribution function is given by
*

*Proposition 3. The -beta distribution satisfies the following basic properties.(i)-beta distribution is the probability distribution that is the area of under a curve is unity.(ii)The mean of this distribution is .(iii)The variance of 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, , is given by
Using the relations (12), (13), and (16), we have

*Proof of (iii). *The variance of is given by
Thus substituting the values of and in (42) along with some algebraic calculations we have the desired result.

*2.2. -Beta Distribution of the Second Kind*

*2.2. -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 .

*Theorem 4. The -beta function of the second kind represents a probability distribution function that is
*

*Proof. *We observe that
Let , so that ; thus by using the relation (11), the above equation gives

*3. Moment Generating Function of -Gamma Distribution*

*3. Moment Generating Function of -Gamma Distribution**In this section, we derive the moment generating function of continuous random variable of newly defined -gamma distribution in terms of a new parameter , which is illustrated as
Let , so that and . Then substituting these values in (48), we obtain
Now differentiating times with respect to and putting , we get
Thus when , we obtain , when , , and hence = variance of the -gamma distribution proved in Proposition 2.*

*3.1. Higher Moment in terms of *

*3.1. Higher Moment in terms of**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
*

*Proof. *Consider
Changing the variables as , above equation becomes
Replacing by , we have
Now using in the above equation we get the desired result.

*4. Conclusion*

*4. 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 , 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 .*

*Conflict of Interests*

*Conflict of Interests**The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgment*

*Acknowledgment**The authors would like to express profound gratitude to referees for deeper review of this paper and the referee’s useful suggestions that led to an improved presentation of the paper.*

*References*

*References*

- E. D. Rainville,
*Special Functions*, The Macmillan, New York, NY, USA, 1960. View at MathSciNet - R. Diaz, and E. Pariguan, “On hypergeometric functions and Pochhammer $k$-symbol,”
*Divulgaciones Matematicas*, vol. 15, no. 2, pp. 179–192, 2007. View at Google Scholar · View at MathSciNet · View at Scopus - M. G. Kendall and A. Stuart,
*The Advanced Theory of Statistics*, vol. 2, Charles Griffin and Company, London, UK, 1961. - R. J. Larsen and M. L. Marx,
*An Introduction to Mathematical Statistics and Its Applications*, Prentice-Hall International, 5th edition, 2011. - C. Walac,
*A Hand Book on Statictical Distributations for Experimentalist*, 2007. - C. G. Kokologiannaki, “Properties and inequalities of generalized $k$-gamma, beta and zeta functions,”
*International Journal of Contemporary Mathematical Sciences*, vol. 5, no. 13–16, pp. 653–660, 2010. View at Google Scholar · View at MathSciNet - C. G. Kokologiannaki and V. Krasniqi, “Some properties of the $k$-gamma function,”
*Le Matematiche*, vol. 68, no. 1, pp. 13–22, 2013. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Krasniqi, “A limit for the {$k$}-gamma and {$k$}-beta function,”
*International Mathematical Forum*, vol. 5, no. 33, pp. 1613–1617, 2010. View at Google Scholar · View at MathSciNet - M. Mansoor, “Determining the
*k*-generalized gamma function ${\Gamma}_{K}\mathcal{X}$ by functional equations,”*Journal of Contemporary Mathematical Sciences*, vol. 4, no. 2, pp. 1037–1042, 2009. View at Google Scholar - S. Mubeen and G. M. Habibullah, “An integral representation of some $k$-hypergeometric functions,”
*International Mathematical Forum*, vol. 7, no. 1–4, pp. 203–207, 2012. View at Google Scholar · View at MathSciNet - S. Mubeen and G. M. Habibullah, “$k$-fractional integrals and application,”
*International Journal of Contemporary Mathematical Sciences*, vol. 7, no. 1–4, pp. 89–94, 2012. View at Google Scholar · View at MathSciNet - S. Mubeen, M. Naz, and G. Rahman, “A note on $k$-hypergemetric differential equations,”
*Journal of Inequalities and Special Functions*, vol. 4, no. 3, pp. 38–43, 2013. View at Google Scholar · View at MathSciNet - S. Mubeen, G. Rahman, A. Rehman, and M. Naz, “Contiguous function relations for $k$-hypergeometric functions,”
*ISRN Mathematical Analysis*, vol. 2014, Article ID 410801, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - S. Mubeen, M. Naz, A. Rehman, and G. Rahman, “Solutions of $k$-hypergeometric differential equations,”
*Journal of Applied Mathematics*, vol. 2014, Article ID 128787, 13 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - F. Merovci, “Power product inequalities for the ${\mathrm{\Gamma}}_{k}$ function,”
*International Journal of Mathematical Analysis*, vol. 4, no. 21, pp. 1007–1012, 2010. View at Google Scholar · View at MathSciNet · View at Scopus - S. Mubeen, A. Rehman, and F. Shaheen, “Properties of $k$-gamma, $k$-beta and $k$-psi functions,”
*Bothalia Journal*, vol. 4, pp. 371–379, 2014. View at Google Scholar - A. Rehman, S. Mubeen, N. Sadiq, and F. Shaheen, “Some inequalities involving
*k*-gamma and*k*-beta functions with applications,”*Journal of Inequalities and Applications*, vol. 2014, article 224, 16 pages, 2014. View at Publisher · View at Google Scholar

*
*