Research ArticleOn 119896-Gamma and 119896-Beta Distributions andMoment 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 gauhar55uomgmailcom
Received 10 February 2014 Revised 29 June 2014 Accepted 4 July 2014 Published 15 July 2014
Academic Editor Chin-Shang Li
Copyright copy 2014 Gauhar Rahman et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The main objective of the present paper is to define 119896-gamma and 119896-beta distributions and moments generating function for thesaid distributions in terms of a new parameter 119896 gt 0 Also the authors prove some properties of these newly defined distributions
1 Basic Definitions
In this section we give some definitions which provide a basefor our main results The definitions (11ndash13) are given in[1] while (14ndash16) are introduced in [2] Also we have takensome statistics related definitions (17ndash111) from [3ndash5]
11 Pochhmmer Symbol Thefactorial function is denoted anddefined by
(119886)119899=
119886 (119886 + 1) (119886 + 2) sdot sdot sdot (119886 + 119899 minus 1) for 119899 ge 1 119886 = 0
1 if 119899 = 0
(1)
The function (119886)119899defined in relation (1) is also known as
Pochhmmer symbol
12 Gamma Function Let 119911 isin C the Euler gamma functionis defined by
Γ (119911) = lim119899rarrinfin
119899119899119911minus1
(119911)119899
(2)
and the integral form of gamma function is given by
Γ (119911) = int
infin
0
119905119911minus1
119890minus119905
119889119905 R (119911) gt 0 (3)
From the relation (3) using integration by parts we can easilyshow that
Γ (119911 + 1) = 119911Γ (119911) (4)The relation between Pochhammer symbol and gammafunction is given by
(119911)119899=Γ (119911 + 119899)
Γ (119911) (5)
13 Beta Function The beta function of two variables isdefined as
119861 (119909 119910) = int
1
0
119905119909minus1
(1 minus 119905)119910minus1
119889119905 Re (119909) Re (119910) gt 0 (6)
and in terms of gamma function it is written as
119861 (119909 119910) =Γ (119909) Γ (119910)
Γ (119909 + 119910) (7)
14 Pochhammer 119896-Symbol For 119896 gt 0 the Pochhammer 119896-symbol is denoted and defined by
(119886)119899119896
= 119886 (119886 + 119896) (119886 + 2119896) sdot sdot sdot (119886 + (119899 minus 1) 119896) for 119899 ge 1 119886 = 0
1 if 119899 = 0
(8)
Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 982013 6 pageshttpdxdoiorg1011552014982013
2 Journal of Probability and Statistics
15 119896-Gamma Function For 119896 gt 0 and 119911 isin C the 119896-gammafunction is defined as
Γ119896(119911) = lim
119899rarrinfin
119899119896119899
(119899119896)119911119896minus1
(119911)119899119896
(9)
and the integral representation of 119896-gamma function is
Γ119896(119911) = int
infin
0
119905119911minus1
119890minus119905119896119896
119889119905 (10)
16 119896-Beta Function For Re(119909)Re(119910) gt 0 the 119896-betafunction of two variables is defined by
119861119896(119909 119910) =
1
119896int
infin
0
119905119909119896minus1
(1 minus 119905)119910119896minus1
119889119905 (11)
and in terms of 119896-gamma function 119896-beta function is definedas
119861119896(119909 119910) =
Γ119896(119909) Γ119896(119910)
Γ119896(119909 + 119910)
(12)
Also the researchers [6ndash10] have worked on the gen-eralized 119896-gamma and 119896-beta functions and discussed thefollowing properties
Γ119896(119909 + 119896) = 119909Γ
119896(119909) (13)
(119909)119899119896
=Γ119896(119909 + 119899119896)
Γ119896(119909)
(14)
Γ119896(119896) = 1 119896 gt 0 (15)
Using the above relations we see that for 119909 119910 gt 0 and 119896 gt
0 the following properties of 119896-beta function are satisfied byauthors (see [6 7 11])
120573119896(119909 + 119896 119910) =
119909
119909 + 119910120573119896(119909 119910) (16)
120573119896(119909 119910 + 119896) =
119910
119909 + 119910120573119896(119909 119910) (17)
120573119896(119909119896 119910119896) =
1
119896120573 (119909 119910) (18)
120573119896(119909 119896) =
1
119909 120573
119896(119896 119910) =
1
119910 (19)
Note that when 119896 rarr 1 120573119896(119909 119910) rarr 120573(119909 119910)
For more details about the theory of 119896-special functionslike 119896-gamma function 119896-beta function 119896-hypergeometricfunctions solutions of 119896-hypergeometric differential equa-tions contegious functions relations inequalities with appli-cations and integral representations with applications involv-ing 119896-gamma and 119896-beta functions and so forth (See [12ndash17])
17 Probability Distribution and Expected Values In a ran-dom experiment with 119899 outcomes suppose a variable 119883
assumes the values 1199091 1199092 1199093 119909
119899with corresponding
probabilities 1198751 1198752 1198753 119875
119899 then this collection is called
probability distribution andΣ119901119894= 1 (in case of discrete distri-
butions) Also if119891(119909) is a continuous probability distributionfunction defined on an interval [119886 119887] then int119887
119886
119891(119909)119889119909 = 1In statistics there are three types of moments which are
(i) moments about any point 119909 = 119886 (ii) moments about119909 = 0 and (iii) moments about mean position of the givendata Also expected value of the variate is defined as the firstmoment of the probability distribution about 119909 = 0 and the119903th moment about mean of the probability distribution isdefined as 119864(119909
119894minus 119909)119903 where 119909 is the mean of the distribution
Also 119864(119909) shows the expected value of the variate 119909 andis defined as the first moment of the probability distributionabout 119909 = 0 that is
1205831015840
1= 119864 (119909) = int
119887
119886
119909119891 (119909) 119889119909 (20)
18 GammaDistribution A continuous random variable119885 issaid to have a gamma distribution with parameter 119898 gt 0 ifits probability distribution function is defined by
119891 (119911) =
1
Γ (119898)119911119898minus1
119890minus119911
0 ≦ 119911 lt infin
0 elsewhere(21)
and its distribution function 119865(119911) is defined by
119865 (119911) =
int
119911
0
1
Γ (119898)119911119898minus1
119890minus119911
119889119911 119911 ge 0
0 119911 lt 0
(22)
which is also called the incomplete gamma function
19 Moment Generating Function of Gamma DistributionThemoment generating function of 119885 is defined by
1198720(119905) = 119864 (119890
119905119885
) = int
infin
0
119890119905119885
119891 (119911) 119889119911
= int
infin
0
1
Γ (119898)119911119898minus1
119890minus119911(1minus119905)
119889119911
(23)
110 Beta Distribution of the First Kind A continuous ran-dom variable 119885 is said to have a beta distribution with twoparameters119898 and 119899 if its probability distribution function isdefined by
119891 (119911) =
1
119861 (119898 119899)119911119898minus1
(1 minus 119911)119899minus1
0 ≦ 119911 ≦ 1 119898 119899 gt 0
0 elsewhere(24)
Journal of Probability and Statistics 3
This distribution is known as a beta distribution of the firstkind and a beta variable of the first kind is referred to as1205731(119898 119899) Its distribution function 119865(119911) is given by
119865 (119911)
=
0 119911 lt 0
int
119911
0
1
119861 (119898 119899)119911119898minus1
(1 minus 119911)119899minus1
119889119911 0 ≦ 119911 ≦ 1 119898 119899 gt 0
0 119911 gt 1
(25)
111 Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891 (119911) =
1
120573 (119898 119899)
119911119898minus1
(1 + 119911)119898+119899
0 ≦ 119911 lt infin 119898 119899 gt 0
0 otherwise(26)
and its probability distribution function is given by
2 Main Results 119896-Gamma and119896-Beta Distributions
In this section we define gamma and beta distributions interms of a new parameter 119896 gt 0 and discuss some propertiesof these distributions in terms of 119896
Definition 1 Let 119885 be a continuous random variable then itis said to have a 119896-gamma distributionwith parameters119898 gt 0
and 119896 gt 0 if its probability density function is defined by
119891119896(119911) =
1
Γ119896(119898)
119911119898minus1
119890minus119911119896119896
0 ≦ 119911 lt infin 119896 gt 0
0 elsewhere(28)
and its distribution function 119865119896(119911) is defined by
119865119896(119911) =
int
119911
0
1
Γ119896(119898)
119911119898minus1
119890minus119911119896119896
119889119911 119911 gt 0
0 119911 lt 0
(29)
Proposition 2 The newly defined Γ119896(119898) distribution satisfies
the following properties
(i) The 119896-gamma distribution is the probability distribu-tion that is area under the curve is unity
(ii) The mean of 119896-gamma distribution is equal to aparameter119898
(iii) The variance of 119896-gamma distribution is equal to theproduct of two parameters119898119896
Proof of (i) Using the definition of 119896-gamma distributionalong with the relation (10) we have
int
infin
0
119891119896(119911) 119889119911 =
1
Γ119896(119898)
int
infin
0
119911119898minus1
119890minus119911119896119896
119889119911 =Γ119896(119898)
Γ119896(119898)
= 1
(30)
Proof of (ii) Asmean of a distribution is the expected value ofthe variate so the mean of the 119896-gamma distribution is givenby
119911 = 119864119896(119885) =
1
Γ119896(119898)
int
infin
0
119911 sdot 119911119898minus1
119890minus119911119896119896
119889119911 (31)
Using the definition of 119896-gamma function and the relation(13) we have
119911 =1
Γ119896(119898)
int
infin
0
119911119898
119890minus119911119896119896
119889119911 =Γ119896(119898 + 119896)
Γ119896(119898)
= 119898Γ119896(119898)
Γ119896(119898)
= 119898
(32)
Proof of (iii) As variance of a distribution is equal to 119864(1199092) minus(119864(119909))
2 so the variance of 119896-gamma distribution is calculatedas
Var119896(119885) = 119864
119896(1198852
) minus (119864119896(119885))2
(33)
Now we have to find 119864119896(1198852
) which is given by
119864119896(1198852
) =1
Γ119896(119898)
int
infin
0
1199112
sdot 119911119898minus1
119890minus119911119896119896
119889119911
=1
Γ119896(119898)
int
infin
0
119911119898+1
119890minus119911119896119896
119889119911
=Γ119896(119898 + 2119896)
Γ119896(119898)
=(119898 + 119896)119898Γ
119896(119898)
Γ119896(119898)
= 119898 (119898 + 119896)
(34)
Thus we obtain the variance of 119896-gamma distribution as
1205902
119896= 119898 (119898 + 119896) minus 119898
2
= 119898119896 (35)
where 1205902119896is the notation of variance present in the literature
21 119896-Beta Distribution of First Kind Let 119885 be a continuousrandom variable then it is said to have a 119896-beta distributionof the first kindwith two parameters119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896119861119896(119898 119899)
119911119898119896minus1
(1 minus 119911)119899119896minus1
0 ≦ 119911 ≦ 1 119898 119899 119896 gt 0
0 elsewhere(36)
4 Journal of Probability and Statistics
In the above distribution the beta variable of the first kind isreferred to as 120573
1119896(119898 119899) and its distribution function 119865
119896(119911) is
given by
119865119896(119911) =
0 119911 lt 0
int
119911
0
1
119896119861119896(119898 119899)
119911119898119896minus1
(1 minus 119911)119899119896minus1
119889119911 0 ≦ 119911 ≦ 1
119898 119899 gt 0
0 119911 gt 1
(37)
Proposition 3 The 119896-beta distribution 1205731119896(119898 119899) satisfies the
following basic properties
(i) 119896-beta distribution is the probability distribution thatis the area of 120573
1119896(119898 119899) under a curve 119891
119896(119911) is unity
(ii) The mean of this distribution is119898(119898 + 119899)(iii) The variance of 120573
Thus substituting the values of119864119896(1198852
) and119864119896(119885) in (42) along
with some algebraic calculations we have the desired result
22 119896-Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a 119896-beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
0 ≦ 119911 lt infin 119898 119899 119896 gt 0
0 otherwise(44)
Note The 119896-beta distribution of the second kind is denotedby 1205732119896(119898 119899)
Theorem 4 The 119896-beta function of the second kind representsa probability distribution function that is
int
infin
0
119891119896(119911) 119889119911 = 1 (45)
Proof We observe that
int
infin
0
119891119896(119911) 119889119911 = int
infin
0
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
119889119911 (46)
Let 1 + 119911 = 1119910 so that 119889119911 = minus1198891199101199102 thus by using the
relation (11) the above equation gives
=1
120573119896(119898 119899)
1
119896int
1
0
119910119899119896minus1
(1 minus 119910)119898119896minus1
119889119910 =120573119896(119898 119899)
120573119896(119898 119899)
= 1
(47)
3 Moment Generating Function of119896-Gamma Distribution
In this section we derive the moment generating functionof continuous random variable 119885 of newly defined 119896-gamma
Journal of Probability and Statistics 5
distribution in terms of a new parameter 119896 gt 0 which isillustrated as
119896(119899)(119899 minus 119896)(119899 minus 2119896) sdot sdot sdot (119899 minus 119903119896) in
the above equation we get the desired result
4 Conclusion
In this paper the authors conclude that we have the following
(i) If 119896 tends to 1 then 119896-gamma distribution and 119896-beta distribution tend to classical gamma and betadistribution
(ii) The authors also conclude that the area of 119896-gammadistribution and 119896-beta distribution for each positivevalue of 119896 is one and theirmean is equal to a parameter119898 and 119898(119898 + 119899) respectively The variance of 119896-gamma distribution for each positive value of 119896 isequal to 119896 times of the parameter 119898 In this case if119896 = 1 then it will be equal to variance of gammadistribution The variance of 119896-beta distribution foreach positive value of 119896 is also defined
(iii) In this paper the authors introduced moments gener-ating function and higher moments in terms of a newparameter 119896 gt 0
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to express profound gratitude toreferees for deeper review of this paper and the refereersquos usefulsuggestions that led to an improved presentation of the paper
References
[1] E D Rainville Special Functions The Macmillan New YorkNY USA 1960
6 Journal of Probability and Statistics
[2] R Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer 119896-symbolrdquoDivulgacionesMatematicas vol 15 no2 pp 179ndash192 2007
[3] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 Charles Griffin and Company London UK 1961
[4] R J Larsen and M L Marx An Introduction to MathematicalStatistics and Its Applications Prentice-Hall International 5thedition 2011
[5] C Walac A Hand Book on Statictical Distributations for Exper-imentalist 2007
[6] C G Kokologiannaki ldquoProperties and inequalities of general-ized 119896-gamma beta and zeta functionsrdquo International Journal ofContemporary Mathematical Sciences vol 5 no 13ndash16 pp 653ndash660 2010
[7] C G Kokologiannaki and V Krasniqi ldquoSome properties of the119896-gamma functionrdquo Le Matematiche vol 68 no 1 pp 13ndash222013
[8] V Krasniqi ldquoA limit for the $k$-gamma and $k$-beta functionrdquoInternational Mathematical Forum vol 5 no 33 pp 1613ndash16172010
[9] M Mansoor ldquoDetermining the k-generalized gamma functionΓ119870X by functional equationsrdquo Journal of Contemporary Mathe-
matical Sciences vol 4 no 2 pp 1037ndash1042 2009[10] S Mubeen andGM Habibullah ldquoAn integral representation of
some 119896-hypergeometric functionsrdquo International MathematicalForum vol 7 no 1ndash4 pp 203ndash207 2012
[11] S Mubeen and G M Habibullah ldquo119896-fractional integrals andapplicationrdquo International Journal of Contemporary Mathemat-ical Sciences vol 7 no 1ndash4 pp 89ndash94 2012
[12] S Mubeen M Naz and G Rahman ldquoA note on 119896-hyper-gemetric differential equationsrdquo Journal of Inequalities andSpecial Functions vol 4 no 3 pp 38ndash43 2013
[13] S Mubeen G Rahman A Rehman and M Naz ldquoContiguousfunction relations for 119896-hypergeometric functionsrdquo ISRNMath-ematical Analysis vol 2014 Article ID 410801 6 pages 2014
[14] S Mubeen M Naz A Rehman and G Rahman ldquoSolutionsof 119896-hypergeometric differential equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 128787 13 pages 2014
[15] F Merovci ldquoPower product inequalities for the Γ119896functionrdquo
International Journal ofMathematical Analysis vol 4 no 21 pp1007ndash1012 2010
[16] S Mubeen A Rehman and F Shaheen ldquoProperties of 119896-gamma 119896-beta and 119896-psi functionsrdquo Bothalia Journal vol 4 pp371ndash379 2014
[17] A Rehman S Mubeen N Sadiq and F Shaheen ldquoSomeinequalities involving k-gamma and k-beta functions withapplicationsrdquo Journal of Inequalities and Applications vol 2014article 224 16 pages 2014
15 119896-Gamma Function For 119896 gt 0 and 119911 isin C the 119896-gammafunction is defined as
Γ119896(119911) = lim
119899rarrinfin
119899119896119899
(119899119896)119911119896minus1
(119911)119899119896
(9)
and the integral representation of 119896-gamma function is
Γ119896(119911) = int
infin
0
119905119911minus1
119890minus119905119896119896
119889119905 (10)
16 119896-Beta Function For Re(119909)Re(119910) gt 0 the 119896-betafunction of two variables is defined by
119861119896(119909 119910) =
1
119896int
infin
0
119905119909119896minus1
(1 minus 119905)119910119896minus1
119889119905 (11)
and in terms of 119896-gamma function 119896-beta function is definedas
119861119896(119909 119910) =
Γ119896(119909) Γ119896(119910)
Γ119896(119909 + 119910)
(12)
Also the researchers [6ndash10] have worked on the gen-eralized 119896-gamma and 119896-beta functions and discussed thefollowing properties
Γ119896(119909 + 119896) = 119909Γ
119896(119909) (13)
(119909)119899119896
=Γ119896(119909 + 119899119896)
Γ119896(119909)
(14)
Γ119896(119896) = 1 119896 gt 0 (15)
Using the above relations we see that for 119909 119910 gt 0 and 119896 gt
0 the following properties of 119896-beta function are satisfied byauthors (see [6 7 11])
120573119896(119909 + 119896 119910) =
119909
119909 + 119910120573119896(119909 119910) (16)
120573119896(119909 119910 + 119896) =
119910
119909 + 119910120573119896(119909 119910) (17)
120573119896(119909119896 119910119896) =
1
119896120573 (119909 119910) (18)
120573119896(119909 119896) =
1
119909 120573
119896(119896 119910) =
1
119910 (19)
Note that when 119896 rarr 1 120573119896(119909 119910) rarr 120573(119909 119910)
For more details about the theory of 119896-special functionslike 119896-gamma function 119896-beta function 119896-hypergeometricfunctions solutions of 119896-hypergeometric differential equa-tions contegious functions relations inequalities with appli-cations and integral representations with applications involv-ing 119896-gamma and 119896-beta functions and so forth (See [12ndash17])
17 Probability Distribution and Expected Values In a ran-dom experiment with 119899 outcomes suppose a variable 119883
assumes the values 1199091 1199092 1199093 119909
119899with corresponding
probabilities 1198751 1198752 1198753 119875
119899 then this collection is called
probability distribution andΣ119901119894= 1 (in case of discrete distri-
butions) Also if119891(119909) is a continuous probability distributionfunction defined on an interval [119886 119887] then int119887
119886
119891(119909)119889119909 = 1In statistics there are three types of moments which are
(i) moments about any point 119909 = 119886 (ii) moments about119909 = 0 and (iii) moments about mean position of the givendata Also expected value of the variate is defined as the firstmoment of the probability distribution about 119909 = 0 and the119903th moment about mean of the probability distribution isdefined as 119864(119909
119894minus 119909)119903 where 119909 is the mean of the distribution
Also 119864(119909) shows the expected value of the variate 119909 andis defined as the first moment of the probability distributionabout 119909 = 0 that is
1205831015840
1= 119864 (119909) = int
119887
119886
119909119891 (119909) 119889119909 (20)
18 GammaDistribution A continuous random variable119885 issaid to have a gamma distribution with parameter 119898 gt 0 ifits probability distribution function is defined by
119891 (119911) =
1
Γ (119898)119911119898minus1
119890minus119911
0 ≦ 119911 lt infin
0 elsewhere(21)
and its distribution function 119865(119911) is defined by
119865 (119911) =
int
119911
0
1
Γ (119898)119911119898minus1
119890minus119911
119889119911 119911 ge 0
0 119911 lt 0
(22)
which is also called the incomplete gamma function
19 Moment Generating Function of Gamma DistributionThemoment generating function of 119885 is defined by
1198720(119905) = 119864 (119890
119905119885
) = int
infin
0
119890119905119885
119891 (119911) 119889119911
= int
infin
0
1
Γ (119898)119911119898minus1
119890minus119911(1minus119905)
119889119911
(23)
110 Beta Distribution of the First Kind A continuous ran-dom variable 119885 is said to have a beta distribution with twoparameters119898 and 119899 if its probability distribution function isdefined by
119891 (119911) =
1
119861 (119898 119899)119911119898minus1
(1 minus 119911)119899minus1
0 ≦ 119911 ≦ 1 119898 119899 gt 0
0 elsewhere(24)
Journal of Probability and Statistics 3
This distribution is known as a beta distribution of the firstkind and a beta variable of the first kind is referred to as1205731(119898 119899) Its distribution function 119865(119911) is given by
119865 (119911)
=
0 119911 lt 0
int
119911
0
1
119861 (119898 119899)119911119898minus1
(1 minus 119911)119899minus1
119889119911 0 ≦ 119911 ≦ 1 119898 119899 gt 0
0 119911 gt 1
(25)
111 Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891 (119911) =
1
120573 (119898 119899)
119911119898minus1
(1 + 119911)119898+119899
0 ≦ 119911 lt infin 119898 119899 gt 0
0 otherwise(26)
and its probability distribution function is given by
2 Main Results 119896-Gamma and119896-Beta Distributions
In this section we define gamma and beta distributions interms of a new parameter 119896 gt 0 and discuss some propertiesof these distributions in terms of 119896
Definition 1 Let 119885 be a continuous random variable then itis said to have a 119896-gamma distributionwith parameters119898 gt 0
and 119896 gt 0 if its probability density function is defined by
119891119896(119911) =
1
Γ119896(119898)
119911119898minus1
119890minus119911119896119896
0 ≦ 119911 lt infin 119896 gt 0
0 elsewhere(28)
and its distribution function 119865119896(119911) is defined by
119865119896(119911) =
int
119911
0
1
Γ119896(119898)
119911119898minus1
119890minus119911119896119896
119889119911 119911 gt 0
0 119911 lt 0
(29)
Proposition 2 The newly defined Γ119896(119898) distribution satisfies
the following properties
(i) The 119896-gamma distribution is the probability distribu-tion that is area under the curve is unity
(ii) The mean of 119896-gamma distribution is equal to aparameter119898
(iii) The variance of 119896-gamma distribution is equal to theproduct of two parameters119898119896
Proof of (i) Using the definition of 119896-gamma distributionalong with the relation (10) we have
int
infin
0
119891119896(119911) 119889119911 =
1
Γ119896(119898)
int
infin
0
119911119898minus1
119890minus119911119896119896
119889119911 =Γ119896(119898)
Γ119896(119898)
= 1
(30)
Proof of (ii) Asmean of a distribution is the expected value ofthe variate so the mean of the 119896-gamma distribution is givenby
119911 = 119864119896(119885) =
1
Γ119896(119898)
int
infin
0
119911 sdot 119911119898minus1
119890minus119911119896119896
119889119911 (31)
Using the definition of 119896-gamma function and the relation(13) we have
119911 =1
Γ119896(119898)
int
infin
0
119911119898
119890minus119911119896119896
119889119911 =Γ119896(119898 + 119896)
Γ119896(119898)
= 119898Γ119896(119898)
Γ119896(119898)
= 119898
(32)
Proof of (iii) As variance of a distribution is equal to 119864(1199092) minus(119864(119909))
2 so the variance of 119896-gamma distribution is calculatedas
Var119896(119885) = 119864
119896(1198852
) minus (119864119896(119885))2
(33)
Now we have to find 119864119896(1198852
) which is given by
119864119896(1198852
) =1
Γ119896(119898)
int
infin
0
1199112
sdot 119911119898minus1
119890minus119911119896119896
119889119911
=1
Γ119896(119898)
int
infin
0
119911119898+1
119890minus119911119896119896
119889119911
=Γ119896(119898 + 2119896)
Γ119896(119898)
=(119898 + 119896)119898Γ
119896(119898)
Γ119896(119898)
= 119898 (119898 + 119896)
(34)
Thus we obtain the variance of 119896-gamma distribution as
1205902
119896= 119898 (119898 + 119896) minus 119898
2
= 119898119896 (35)
where 1205902119896is the notation of variance present in the literature
21 119896-Beta Distribution of First Kind Let 119885 be a continuousrandom variable then it is said to have a 119896-beta distributionof the first kindwith two parameters119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896119861119896(119898 119899)
119911119898119896minus1
(1 minus 119911)119899119896minus1
0 ≦ 119911 ≦ 1 119898 119899 119896 gt 0
0 elsewhere(36)
4 Journal of Probability and Statistics
In the above distribution the beta variable of the first kind isreferred to as 120573
1119896(119898 119899) and its distribution function 119865
119896(119911) is
given by
119865119896(119911) =
0 119911 lt 0
int
119911
0
1
119896119861119896(119898 119899)
119911119898119896minus1
(1 minus 119911)119899119896minus1
119889119911 0 ≦ 119911 ≦ 1
119898 119899 gt 0
0 119911 gt 1
(37)
Proposition 3 The 119896-beta distribution 1205731119896(119898 119899) satisfies the
following basic properties
(i) 119896-beta distribution is the probability distribution thatis the area of 120573
1119896(119898 119899) under a curve 119891
119896(119911) is unity
(ii) The mean of this distribution is119898(119898 + 119899)(iii) The variance of 120573
Thus substituting the values of119864119896(1198852
) and119864119896(119885) in (42) along
with some algebraic calculations we have the desired result
22 119896-Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a 119896-beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
0 ≦ 119911 lt infin 119898 119899 119896 gt 0
0 otherwise(44)
Note The 119896-beta distribution of the second kind is denotedby 1205732119896(119898 119899)
Theorem 4 The 119896-beta function of the second kind representsa probability distribution function that is
int
infin
0
119891119896(119911) 119889119911 = 1 (45)
Proof We observe that
int
infin
0
119891119896(119911) 119889119911 = int
infin
0
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
119889119911 (46)
Let 1 + 119911 = 1119910 so that 119889119911 = minus1198891199101199102 thus by using the
relation (11) the above equation gives
=1
120573119896(119898 119899)
1
119896int
1
0
119910119899119896minus1
(1 minus 119910)119898119896minus1
119889119910 =120573119896(119898 119899)
120573119896(119898 119899)
= 1
(47)
3 Moment Generating Function of119896-Gamma Distribution
In this section we derive the moment generating functionof continuous random variable 119885 of newly defined 119896-gamma
Journal of Probability and Statistics 5
distribution in terms of a new parameter 119896 gt 0 which isillustrated as
119896(119899)(119899 minus 119896)(119899 minus 2119896) sdot sdot sdot (119899 minus 119903119896) in
the above equation we get the desired result
4 Conclusion
In this paper the authors conclude that we have the following
(i) If 119896 tends to 1 then 119896-gamma distribution and 119896-beta distribution tend to classical gamma and betadistribution
(ii) The authors also conclude that the area of 119896-gammadistribution and 119896-beta distribution for each positivevalue of 119896 is one and theirmean is equal to a parameter119898 and 119898(119898 + 119899) respectively The variance of 119896-gamma distribution for each positive value of 119896 isequal to 119896 times of the parameter 119898 In this case if119896 = 1 then it will be equal to variance of gammadistribution The variance of 119896-beta distribution foreach positive value of 119896 is also defined
(iii) In this paper the authors introduced moments gener-ating function and higher moments in terms of a newparameter 119896 gt 0
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to express profound gratitude toreferees for deeper review of this paper and the refereersquos usefulsuggestions that led to an improved presentation of the paper
References
[1] E D Rainville Special Functions The Macmillan New YorkNY USA 1960
6 Journal of Probability and Statistics
[2] R Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer 119896-symbolrdquoDivulgacionesMatematicas vol 15 no2 pp 179ndash192 2007
[3] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 Charles Griffin and Company London UK 1961
[4] R J Larsen and M L Marx An Introduction to MathematicalStatistics and Its Applications Prentice-Hall International 5thedition 2011
[5] C Walac A Hand Book on Statictical Distributations for Exper-imentalist 2007
[6] C G Kokologiannaki ldquoProperties and inequalities of general-ized 119896-gamma beta and zeta functionsrdquo International Journal ofContemporary Mathematical Sciences vol 5 no 13ndash16 pp 653ndash660 2010
[7] C G Kokologiannaki and V Krasniqi ldquoSome properties of the119896-gamma functionrdquo Le Matematiche vol 68 no 1 pp 13ndash222013
[8] V Krasniqi ldquoA limit for the $k$-gamma and $k$-beta functionrdquoInternational Mathematical Forum vol 5 no 33 pp 1613ndash16172010
[9] M Mansoor ldquoDetermining the k-generalized gamma functionΓ119870X by functional equationsrdquo Journal of Contemporary Mathe-
matical Sciences vol 4 no 2 pp 1037ndash1042 2009[10] S Mubeen andGM Habibullah ldquoAn integral representation of
some 119896-hypergeometric functionsrdquo International MathematicalForum vol 7 no 1ndash4 pp 203ndash207 2012
[11] S Mubeen and G M Habibullah ldquo119896-fractional integrals andapplicationrdquo International Journal of Contemporary Mathemat-ical Sciences vol 7 no 1ndash4 pp 89ndash94 2012
[12] S Mubeen M Naz and G Rahman ldquoA note on 119896-hyper-gemetric differential equationsrdquo Journal of Inequalities andSpecial Functions vol 4 no 3 pp 38ndash43 2013
[13] S Mubeen G Rahman A Rehman and M Naz ldquoContiguousfunction relations for 119896-hypergeometric functionsrdquo ISRNMath-ematical Analysis vol 2014 Article ID 410801 6 pages 2014
[14] S Mubeen M Naz A Rehman and G Rahman ldquoSolutionsof 119896-hypergeometric differential equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 128787 13 pages 2014
[15] F Merovci ldquoPower product inequalities for the Γ119896functionrdquo
International Journal ofMathematical Analysis vol 4 no 21 pp1007ndash1012 2010
[16] S Mubeen A Rehman and F Shaheen ldquoProperties of 119896-gamma 119896-beta and 119896-psi functionsrdquo Bothalia Journal vol 4 pp371ndash379 2014
[17] A Rehman S Mubeen N Sadiq and F Shaheen ldquoSomeinequalities involving k-gamma and k-beta functions withapplicationsrdquo Journal of Inequalities and Applications vol 2014article 224 16 pages 2014
This distribution is known as a beta distribution of the firstkind and a beta variable of the first kind is referred to as1205731(119898 119899) Its distribution function 119865(119911) is given by
119865 (119911)
=
0 119911 lt 0
int
119911
0
1
119861 (119898 119899)119911119898minus1
(1 minus 119911)119899minus1
119889119911 0 ≦ 119911 ≦ 1 119898 119899 gt 0
0 119911 gt 1
(25)
111 Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891 (119911) =
1
120573 (119898 119899)
119911119898minus1
(1 + 119911)119898+119899
0 ≦ 119911 lt infin 119898 119899 gt 0
0 otherwise(26)
and its probability distribution function is given by
2 Main Results 119896-Gamma and119896-Beta Distributions
In this section we define gamma and beta distributions interms of a new parameter 119896 gt 0 and discuss some propertiesof these distributions in terms of 119896
Definition 1 Let 119885 be a continuous random variable then itis said to have a 119896-gamma distributionwith parameters119898 gt 0
and 119896 gt 0 if its probability density function is defined by
119891119896(119911) =
1
Γ119896(119898)
119911119898minus1
119890minus119911119896119896
0 ≦ 119911 lt infin 119896 gt 0
0 elsewhere(28)
and its distribution function 119865119896(119911) is defined by
119865119896(119911) =
int
119911
0
1
Γ119896(119898)
119911119898minus1
119890minus119911119896119896
119889119911 119911 gt 0
0 119911 lt 0
(29)
Proposition 2 The newly defined Γ119896(119898) distribution satisfies
the following properties
(i) The 119896-gamma distribution is the probability distribu-tion that is area under the curve is unity
(ii) The mean of 119896-gamma distribution is equal to aparameter119898
(iii) The variance of 119896-gamma distribution is equal to theproduct of two parameters119898119896
Proof of (i) Using the definition of 119896-gamma distributionalong with the relation (10) we have
int
infin
0
119891119896(119911) 119889119911 =
1
Γ119896(119898)
int
infin
0
119911119898minus1
119890minus119911119896119896
119889119911 =Γ119896(119898)
Γ119896(119898)
= 1
(30)
Proof of (ii) Asmean of a distribution is the expected value ofthe variate so the mean of the 119896-gamma distribution is givenby
119911 = 119864119896(119885) =
1
Γ119896(119898)
int
infin
0
119911 sdot 119911119898minus1
119890minus119911119896119896
119889119911 (31)
Using the definition of 119896-gamma function and the relation(13) we have
119911 =1
Γ119896(119898)
int
infin
0
119911119898
119890minus119911119896119896
119889119911 =Γ119896(119898 + 119896)
Γ119896(119898)
= 119898Γ119896(119898)
Γ119896(119898)
= 119898
(32)
Proof of (iii) As variance of a distribution is equal to 119864(1199092) minus(119864(119909))
2 so the variance of 119896-gamma distribution is calculatedas
Var119896(119885) = 119864
119896(1198852
) minus (119864119896(119885))2
(33)
Now we have to find 119864119896(1198852
) which is given by
119864119896(1198852
) =1
Γ119896(119898)
int
infin
0
1199112
sdot 119911119898minus1
119890minus119911119896119896
119889119911
=1
Γ119896(119898)
int
infin
0
119911119898+1
119890minus119911119896119896
119889119911
=Γ119896(119898 + 2119896)
Γ119896(119898)
=(119898 + 119896)119898Γ
119896(119898)
Γ119896(119898)
= 119898 (119898 + 119896)
(34)
Thus we obtain the variance of 119896-gamma distribution as
1205902
119896= 119898 (119898 + 119896) minus 119898
2
= 119898119896 (35)
where 1205902119896is the notation of variance present in the literature
21 119896-Beta Distribution of First Kind Let 119885 be a continuousrandom variable then it is said to have a 119896-beta distributionof the first kindwith two parameters119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896119861119896(119898 119899)
119911119898119896minus1
(1 minus 119911)119899119896minus1
0 ≦ 119911 ≦ 1 119898 119899 119896 gt 0
0 elsewhere(36)
4 Journal of Probability and Statistics
In the above distribution the beta variable of the first kind isreferred to as 120573
1119896(119898 119899) and its distribution function 119865
119896(119911) is
given by
119865119896(119911) =
0 119911 lt 0
int
119911
0
1
119896119861119896(119898 119899)
119911119898119896minus1
(1 minus 119911)119899119896minus1
119889119911 0 ≦ 119911 ≦ 1
119898 119899 gt 0
0 119911 gt 1
(37)
Proposition 3 The 119896-beta distribution 1205731119896(119898 119899) satisfies the
following basic properties
(i) 119896-beta distribution is the probability distribution thatis the area of 120573
1119896(119898 119899) under a curve 119891
119896(119911) is unity
(ii) The mean of this distribution is119898(119898 + 119899)(iii) The variance of 120573
Thus substituting the values of119864119896(1198852
) and119864119896(119885) in (42) along
with some algebraic calculations we have the desired result
22 119896-Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a 119896-beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
0 ≦ 119911 lt infin 119898 119899 119896 gt 0
0 otherwise(44)
Note The 119896-beta distribution of the second kind is denotedby 1205732119896(119898 119899)
Theorem 4 The 119896-beta function of the second kind representsa probability distribution function that is
int
infin
0
119891119896(119911) 119889119911 = 1 (45)
Proof We observe that
int
infin
0
119891119896(119911) 119889119911 = int
infin
0
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
119889119911 (46)
Let 1 + 119911 = 1119910 so that 119889119911 = minus1198891199101199102 thus by using the
relation (11) the above equation gives
=1
120573119896(119898 119899)
1
119896int
1
0
119910119899119896minus1
(1 minus 119910)119898119896minus1
119889119910 =120573119896(119898 119899)
120573119896(119898 119899)
= 1
(47)
3 Moment Generating Function of119896-Gamma Distribution
In this section we derive the moment generating functionof continuous random variable 119885 of newly defined 119896-gamma
Journal of Probability and Statistics 5
distribution in terms of a new parameter 119896 gt 0 which isillustrated as
119896(119899)(119899 minus 119896)(119899 minus 2119896) sdot sdot sdot (119899 minus 119903119896) in
the above equation we get the desired result
4 Conclusion
In this paper the authors conclude that we have the following
(i) If 119896 tends to 1 then 119896-gamma distribution and 119896-beta distribution tend to classical gamma and betadistribution
(ii) The authors also conclude that the area of 119896-gammadistribution and 119896-beta distribution for each positivevalue of 119896 is one and theirmean is equal to a parameter119898 and 119898(119898 + 119899) respectively The variance of 119896-gamma distribution for each positive value of 119896 isequal to 119896 times of the parameter 119898 In this case if119896 = 1 then it will be equal to variance of gammadistribution The variance of 119896-beta distribution foreach positive value of 119896 is also defined
(iii) In this paper the authors introduced moments gener-ating function and higher moments in terms of a newparameter 119896 gt 0
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to express profound gratitude toreferees for deeper review of this paper and the refereersquos usefulsuggestions that led to an improved presentation of the paper
References
[1] E D Rainville Special Functions The Macmillan New YorkNY USA 1960
6 Journal of Probability and Statistics
[2] R Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer 119896-symbolrdquoDivulgacionesMatematicas vol 15 no2 pp 179ndash192 2007
[3] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 Charles Griffin and Company London UK 1961
[4] R J Larsen and M L Marx An Introduction to MathematicalStatistics and Its Applications Prentice-Hall International 5thedition 2011
[5] C Walac A Hand Book on Statictical Distributations for Exper-imentalist 2007
[6] C G Kokologiannaki ldquoProperties and inequalities of general-ized 119896-gamma beta and zeta functionsrdquo International Journal ofContemporary Mathematical Sciences vol 5 no 13ndash16 pp 653ndash660 2010
[7] C G Kokologiannaki and V Krasniqi ldquoSome properties of the119896-gamma functionrdquo Le Matematiche vol 68 no 1 pp 13ndash222013
[8] V Krasniqi ldquoA limit for the $k$-gamma and $k$-beta functionrdquoInternational Mathematical Forum vol 5 no 33 pp 1613ndash16172010
[9] M Mansoor ldquoDetermining the k-generalized gamma functionΓ119870X by functional equationsrdquo Journal of Contemporary Mathe-
matical Sciences vol 4 no 2 pp 1037ndash1042 2009[10] S Mubeen andGM Habibullah ldquoAn integral representation of
some 119896-hypergeometric functionsrdquo International MathematicalForum vol 7 no 1ndash4 pp 203ndash207 2012
[11] S Mubeen and G M Habibullah ldquo119896-fractional integrals andapplicationrdquo International Journal of Contemporary Mathemat-ical Sciences vol 7 no 1ndash4 pp 89ndash94 2012
[12] S Mubeen M Naz and G Rahman ldquoA note on 119896-hyper-gemetric differential equationsrdquo Journal of Inequalities andSpecial Functions vol 4 no 3 pp 38ndash43 2013
[13] S Mubeen G Rahman A Rehman and M Naz ldquoContiguousfunction relations for 119896-hypergeometric functionsrdquo ISRNMath-ematical Analysis vol 2014 Article ID 410801 6 pages 2014
[14] S Mubeen M Naz A Rehman and G Rahman ldquoSolutionsof 119896-hypergeometric differential equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 128787 13 pages 2014
[15] F Merovci ldquoPower product inequalities for the Γ119896functionrdquo
International Journal ofMathematical Analysis vol 4 no 21 pp1007ndash1012 2010
[16] S Mubeen A Rehman and F Shaheen ldquoProperties of 119896-gamma 119896-beta and 119896-psi functionsrdquo Bothalia Journal vol 4 pp371ndash379 2014
[17] A Rehman S Mubeen N Sadiq and F Shaheen ldquoSomeinequalities involving k-gamma and k-beta functions withapplicationsrdquo Journal of Inequalities and Applications vol 2014article 224 16 pages 2014
Thus substituting the values of119864119896(1198852
) and119864119896(119885) in (42) along
with some algebraic calculations we have the desired result
22 119896-Beta Distribution of the Second Kind A continuousrandom variable 119885 is said to have a 119896-beta distribution ofthe second kind with parameters 119898 and 119899 if its probabilitydistribution function is defined by
119891119896(119911)
=
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
0 ≦ 119911 lt infin 119898 119899 119896 gt 0
0 otherwise(44)
Note The 119896-beta distribution of the second kind is denotedby 1205732119896(119898 119899)
Theorem 4 The 119896-beta function of the second kind representsa probability distribution function that is
int
infin
0
119891119896(119911) 119889119911 = 1 (45)
Proof We observe that
int
infin
0
119891119896(119911) 119889119911 = int
infin
0
1
119896120573119896(119898 119899)
119911119898119896minus1
(1 + 119911)(119898+119899)119896
119889119911 (46)
Let 1 + 119911 = 1119910 so that 119889119911 = minus1198891199101199102 thus by using the
relation (11) the above equation gives
=1
120573119896(119898 119899)
1
119896int
1
0
119910119899119896minus1
(1 minus 119910)119898119896minus1
119889119910 =120573119896(119898 119899)
120573119896(119898 119899)
= 1
(47)
3 Moment Generating Function of119896-Gamma Distribution
In this section we derive the moment generating functionof continuous random variable 119885 of newly defined 119896-gamma
Journal of Probability and Statistics 5
distribution in terms of a new parameter 119896 gt 0 which isillustrated as
119896(119899)(119899 minus 119896)(119899 minus 2119896) sdot sdot sdot (119899 minus 119903119896) in
the above equation we get the desired result
4 Conclusion
In this paper the authors conclude that we have the following
(i) If 119896 tends to 1 then 119896-gamma distribution and 119896-beta distribution tend to classical gamma and betadistribution
(ii) The authors also conclude that the area of 119896-gammadistribution and 119896-beta distribution for each positivevalue of 119896 is one and theirmean is equal to a parameter119898 and 119898(119898 + 119899) respectively The variance of 119896-gamma distribution for each positive value of 119896 isequal to 119896 times of the parameter 119898 In this case if119896 = 1 then it will be equal to variance of gammadistribution The variance of 119896-beta distribution foreach positive value of 119896 is also defined
(iii) In this paper the authors introduced moments gener-ating function and higher moments in terms of a newparameter 119896 gt 0
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to express profound gratitude toreferees for deeper review of this paper and the refereersquos usefulsuggestions that led to an improved presentation of the paper
References
[1] E D Rainville Special Functions The Macmillan New YorkNY USA 1960
6 Journal of Probability and Statistics
[2] R Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer 119896-symbolrdquoDivulgacionesMatematicas vol 15 no2 pp 179ndash192 2007
[3] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 Charles Griffin and Company London UK 1961
[4] R J Larsen and M L Marx An Introduction to MathematicalStatistics and Its Applications Prentice-Hall International 5thedition 2011
[5] C Walac A Hand Book on Statictical Distributations for Exper-imentalist 2007
[6] C G Kokologiannaki ldquoProperties and inequalities of general-ized 119896-gamma beta and zeta functionsrdquo International Journal ofContemporary Mathematical Sciences vol 5 no 13ndash16 pp 653ndash660 2010
[7] C G Kokologiannaki and V Krasniqi ldquoSome properties of the119896-gamma functionrdquo Le Matematiche vol 68 no 1 pp 13ndash222013
[8] V Krasniqi ldquoA limit for the $k$-gamma and $k$-beta functionrdquoInternational Mathematical Forum vol 5 no 33 pp 1613ndash16172010
[9] M Mansoor ldquoDetermining the k-generalized gamma functionΓ119870X by functional equationsrdquo Journal of Contemporary Mathe-
matical Sciences vol 4 no 2 pp 1037ndash1042 2009[10] S Mubeen andGM Habibullah ldquoAn integral representation of
some 119896-hypergeometric functionsrdquo International MathematicalForum vol 7 no 1ndash4 pp 203ndash207 2012
[11] S Mubeen and G M Habibullah ldquo119896-fractional integrals andapplicationrdquo International Journal of Contemporary Mathemat-ical Sciences vol 7 no 1ndash4 pp 89ndash94 2012
[12] S Mubeen M Naz and G Rahman ldquoA note on 119896-hyper-gemetric differential equationsrdquo Journal of Inequalities andSpecial Functions vol 4 no 3 pp 38ndash43 2013
[13] S Mubeen G Rahman A Rehman and M Naz ldquoContiguousfunction relations for 119896-hypergeometric functionsrdquo ISRNMath-ematical Analysis vol 2014 Article ID 410801 6 pages 2014
[14] S Mubeen M Naz A Rehman and G Rahman ldquoSolutionsof 119896-hypergeometric differential equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 128787 13 pages 2014
[15] F Merovci ldquoPower product inequalities for the Γ119896functionrdquo
International Journal ofMathematical Analysis vol 4 no 21 pp1007ndash1012 2010
[16] S Mubeen A Rehman and F Shaheen ldquoProperties of 119896-gamma 119896-beta and 119896-psi functionsrdquo Bothalia Journal vol 4 pp371ndash379 2014
[17] A Rehman S Mubeen N Sadiq and F Shaheen ldquoSomeinequalities involving k-gamma and k-beta functions withapplicationsrdquo Journal of Inequalities and Applications vol 2014article 224 16 pages 2014
119896(119899)(119899 minus 119896)(119899 minus 2119896) sdot sdot sdot (119899 minus 119903119896) in
the above equation we get the desired result
4 Conclusion
In this paper the authors conclude that we have the following
(i) If 119896 tends to 1 then 119896-gamma distribution and 119896-beta distribution tend to classical gamma and betadistribution
(ii) The authors also conclude that the area of 119896-gammadistribution and 119896-beta distribution for each positivevalue of 119896 is one and theirmean is equal to a parameter119898 and 119898(119898 + 119899) respectively The variance of 119896-gamma distribution for each positive value of 119896 isequal to 119896 times of the parameter 119898 In this case if119896 = 1 then it will be equal to variance of gammadistribution The variance of 119896-beta distribution foreach positive value of 119896 is also defined
(iii) In this paper the authors introduced moments gener-ating function and higher moments in terms of a newparameter 119896 gt 0
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to express profound gratitude toreferees for deeper review of this paper and the refereersquos usefulsuggestions that led to an improved presentation of the paper
References
[1] E D Rainville Special Functions The Macmillan New YorkNY USA 1960
6 Journal of Probability and Statistics
[2] R Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer 119896-symbolrdquoDivulgacionesMatematicas vol 15 no2 pp 179ndash192 2007
[3] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 Charles Griffin and Company London UK 1961
[4] R J Larsen and M L Marx An Introduction to MathematicalStatistics and Its Applications Prentice-Hall International 5thedition 2011
[5] C Walac A Hand Book on Statictical Distributations for Exper-imentalist 2007
[6] C G Kokologiannaki ldquoProperties and inequalities of general-ized 119896-gamma beta and zeta functionsrdquo International Journal ofContemporary Mathematical Sciences vol 5 no 13ndash16 pp 653ndash660 2010
[7] C G Kokologiannaki and V Krasniqi ldquoSome properties of the119896-gamma functionrdquo Le Matematiche vol 68 no 1 pp 13ndash222013
[8] V Krasniqi ldquoA limit for the $k$-gamma and $k$-beta functionrdquoInternational Mathematical Forum vol 5 no 33 pp 1613ndash16172010
[9] M Mansoor ldquoDetermining the k-generalized gamma functionΓ119870X by functional equationsrdquo Journal of Contemporary Mathe-
matical Sciences vol 4 no 2 pp 1037ndash1042 2009[10] S Mubeen andGM Habibullah ldquoAn integral representation of
some 119896-hypergeometric functionsrdquo International MathematicalForum vol 7 no 1ndash4 pp 203ndash207 2012
[11] S Mubeen and G M Habibullah ldquo119896-fractional integrals andapplicationrdquo International Journal of Contemporary Mathemat-ical Sciences vol 7 no 1ndash4 pp 89ndash94 2012
[12] S Mubeen M Naz and G Rahman ldquoA note on 119896-hyper-gemetric differential equationsrdquo Journal of Inequalities andSpecial Functions vol 4 no 3 pp 38ndash43 2013
[13] S Mubeen G Rahman A Rehman and M Naz ldquoContiguousfunction relations for 119896-hypergeometric functionsrdquo ISRNMath-ematical Analysis vol 2014 Article ID 410801 6 pages 2014
[14] S Mubeen M Naz A Rehman and G Rahman ldquoSolutionsof 119896-hypergeometric differential equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 128787 13 pages 2014
[15] F Merovci ldquoPower product inequalities for the Γ119896functionrdquo
International Journal ofMathematical Analysis vol 4 no 21 pp1007ndash1012 2010
[16] S Mubeen A Rehman and F Shaheen ldquoProperties of 119896-gamma 119896-beta and 119896-psi functionsrdquo Bothalia Journal vol 4 pp371ndash379 2014
[17] A Rehman S Mubeen N Sadiq and F Shaheen ldquoSomeinequalities involving k-gamma and k-beta functions withapplicationsrdquo Journal of Inequalities and Applications vol 2014article 224 16 pages 2014
[2] R Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer 119896-symbolrdquoDivulgacionesMatematicas vol 15 no2 pp 179ndash192 2007
[3] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 Charles Griffin and Company London UK 1961
[4] R J Larsen and M L Marx An Introduction to MathematicalStatistics and Its Applications Prentice-Hall International 5thedition 2011
[5] C Walac A Hand Book on Statictical Distributations for Exper-imentalist 2007
[6] C G Kokologiannaki ldquoProperties and inequalities of general-ized 119896-gamma beta and zeta functionsrdquo International Journal ofContemporary Mathematical Sciences vol 5 no 13ndash16 pp 653ndash660 2010
[7] C G Kokologiannaki and V Krasniqi ldquoSome properties of the119896-gamma functionrdquo Le Matematiche vol 68 no 1 pp 13ndash222013
[8] V Krasniqi ldquoA limit for the $k$-gamma and $k$-beta functionrdquoInternational Mathematical Forum vol 5 no 33 pp 1613ndash16172010
[9] M Mansoor ldquoDetermining the k-generalized gamma functionΓ119870X by functional equationsrdquo Journal of Contemporary Mathe-
matical Sciences vol 4 no 2 pp 1037ndash1042 2009[10] S Mubeen andGM Habibullah ldquoAn integral representation of
some 119896-hypergeometric functionsrdquo International MathematicalForum vol 7 no 1ndash4 pp 203ndash207 2012
[11] S Mubeen and G M Habibullah ldquo119896-fractional integrals andapplicationrdquo International Journal of Contemporary Mathemat-ical Sciences vol 7 no 1ndash4 pp 89ndash94 2012
[12] S Mubeen M Naz and G Rahman ldquoA note on 119896-hyper-gemetric differential equationsrdquo Journal of Inequalities andSpecial Functions vol 4 no 3 pp 38ndash43 2013
[13] S Mubeen G Rahman A Rehman and M Naz ldquoContiguousfunction relations for 119896-hypergeometric functionsrdquo ISRNMath-ematical Analysis vol 2014 Article ID 410801 6 pages 2014
[14] S Mubeen M Naz A Rehman and G Rahman ldquoSolutionsof 119896-hypergeometric differential equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 128787 13 pages 2014
[15] F Merovci ldquoPower product inequalities for the Γ119896functionrdquo
International Journal ofMathematical Analysis vol 4 no 21 pp1007ndash1012 2010
[16] S Mubeen A Rehman and F Shaheen ldquoProperties of 119896-gamma 119896-beta and 119896-psi functionsrdquo Bothalia Journal vol 4 pp371ndash379 2014
[17] A Rehman S Mubeen N Sadiq and F Shaheen ldquoSomeinequalities involving k-gamma and k-beta functions withapplicationsrdquo Journal of Inequalities and Applications vol 2014article 224 16 pages 2014
