Research Article
On -Gamma and -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. Tis 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.
Te main objective of the present paper is to defne -gamma and -beta distributions and moments generating function for the
said distributions in terms of a new parameter >0. Also, the authors prove some properties of these newly defned distributions.
1. Basic Definitions
In this section we give some defnitions which provide a base
for our main results. Te defnitions (1.1–1.3) are given in
[1] while (1.4–1.6) are introduced in [2]. Also, we have taken
some statistics related defnitions (1.7–1.11) from [3–5].
1.1. Pochhmmer Symbol. Te factorial function is denoted and
defned by
()
={
(+1)(+2)⋅⋅⋅(+−1); for ≥ 1, ̸ =0,
1; if =0.
(1)
Te function ()
defned in relation (1) is also known as
Pochhmmer symbol.
1.2. Gamma Function. Let ∈ C; the Euler gamma function
is defned by
Γ()= lim
→∞
!
−1
()
(2)
and the integral form of gamma function is given by
Γ()=∫
∞
0
−1
−
, R () > 0. (3)
From the relation (3), using integration by parts, we can easily
show that
Γ(+1)=Γ(). (4)
Te relation between Pochhammer symbol and gamma
function is given by
()
=
Γ(+)
Γ()
. (5)
1.3. Beta Function. Te beta function of two variables is
defned as
(,)=∫
1
0
−1
(1−)
−1
, Re (), Re () > 0 (6)
and, in terms of gamma function, it is written as
(,)=
Γ()Γ ()
Γ(+)
. (7)
1.4. Pochhammer -Symbol. For >0, the Pochhammer -
symbol is denoted and defned by
()
,
={
(+)(+2)⋅⋅⋅(+(−1)); for ≥ 1, ̸ =0,
1; if =0.
(8)
Hindawi Publishing Corporation
Journal of Probability and Statistics
Volume 2014, Article ID 982013, 6 pages
http://dx.doi.org/10.1155/2014/982013