 Open Access
 Total Downloads : 3
 Authors : Dr. Anjali Mathur, Dr. Sandeep Mathur
 Paper ID : IJERTCONV2IS03085
 Volume & Issue : ETRASCT – 2014 (Volume 2 – Issue 03)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A GENERALIZED GAMMA DISTRIBUTION ASSOCIATED WITH pFq AND ITS APPLICATION IN RELIABILITY
International Journal of Engineering Research & Technology (IJERT)
A GENERALIZED GAMMA DISTRIBUTION ASSOCIATED ISSN: 22780181
ETRASCT' 14 Conference Proceedings
WITH pFq AND ITS APPLICATION IN RELIABILITY
Dr. Anjali Mathur1
Associate Professor, Department of Mathematics, Jodhpur Institute of Engg. & Technology, Jodhpur (Rajasthan) Email :
anjali24mathur@gmail.com
Dr. Sandeep Mathur2
Associate Professor, Department of Mathematics, Jodhpur Institute of Engg. & Technology, Jodhpur (Rajasthan)
Email : mathur.sandeep1979@gmail.com
ABSTRACT
In this paper a generalized gamma distribution involving pFq (generalized hypergeometric function), has been studied from which almost variance. Hazard function have been worked out for generalized gamma distribution having pFq.

INTRODUCTION
In recent years many generalizations of gamma and Weibull distributions are proposed notably by Bradley [1], Srivastava [2], Lee and Gross[4], Bondesson [5]. These generalized distributions are mainly introduced in order to extend the scope of ordering gamma and Weibull distributions and to develop a Model for failure to suit any given particular situation.
Kobayashi [3] has introduced a new type of generalized gamma function as
r (m , n) = xm1 (x+n)rex dx … (1.1)
0
For a positive integer r. Here m and n are parameters of the functions. This function occurs in many problems of diffraction theory [Kobayashi [3]]. However, this generalized gamma function has not yet drawn, the attention of statistician. Agarwal and Kalla [6] has introduced a slightly modified form of the generalized gamma function as
xm1 (x n) ebx
0
dx bm
m, bn
… (1.2)
In this paper we defined a new generalization of gamma distribution involving pFq by considering a modified from of the Agarwal and Kalla [6]. A few well known probability distributions are shown to be its particular cases.
ISSN: 22780181
FUNCTION OF MATRIX ARGUMENT IN THE COMIntPerLnaEtioXnal CJouArnSalEof E:ngineering Research & Technology (IJERT)
ETRASCT' 14 C~onference Proceedings
We consider real valued scalar function of a single matrix argument of the type Z =
~ ~ ~ ~
X + i Y where X and Y are p x p matrices with real elements and i
1 as well as
scaler functions of many matrices
~
Z j , j = 1, 2, .K where each
~ ~
Z j is of the type Z
above in the real case. We confined our discussion to the situation where the argument matrix was real symmetric positive definite. This was done so that the fractional power of matrices and functions of such matrices could be uniquely defined. Corresponding properties are of we restrict to the class of Hermitian positive definite matrices.
Definition : Hermitian positive definite matrix due to Mathai [11], We will denote the
~ ~ ~ ~ ~
conjugate of Z by Z if Z hermitian, then Z = Z *, that is
~ ~ ~ ~ ~ ~ ~ ~
Z = Z * X + i Y = ( X + i )* = X + i Y
~ ~ ~ ~
X = X and Y = Y
~ ~ ~
Thus X is the symmetric and Y is skew symmetric. Further if Z is hermitian
~
positive definite, then all the eigen values of gamma in the complex case is
Z are real and positive. Further, matrix variate
p
~ ()
p ( p1) ( )( 1)……. ( 1)
= 2
~
We will use the notation
Z 0
to indicate that
~
Z is hermitian positive definite.
Constant matrices will be written without a tilde whether the elements are real or complex unless it has to be emphasized that the matrix involved has complex elements. Then in that case a constant matrix will also be written with a tilde.
ZONAL POLYNOMIAL
Let
~
Vk be the vector space of homogeneous polynomial of degree k, then the Zonal
polynomial ~ ~ is defined as the component of (tr ~ k ) in the subspace
~ . ~ ~ is
Ck (X)
X Vk Ck (X)
also generalization of
~ k
~
~
X . The exponential function has the following expansion
~
etr( X)
tr~
C X
(1.3)
1
X
k
K
k 0 k! k0 K k!
The binomial expansion is the following for I ~ ~ ~* 0and all eigen values
X 0 that is X X
X
of ~ are between 0 and 1.
~
~ ~
C
International Journal of Engineering Research & Technology (IJERT)
det I X
where
k 0 K
K
k! k
(X),
(1.4)ISSN: 22780181
ETRASCT' 14 Conference Proceedings
()
p
j 1 ,
2
K
j
j1 k
with
K k ,…, k , k … k k
1 p 1 p
~ ~ ~ ~
~ ~ ~ ~ ~ ~
CK (X)CK (T)
CK
CK (H * XHT)dH
~ O ( P )
~ (I)
(1.5)
where I is the identity matrix, the integral is over the orthogonal group of p x p matrices and
d ~
H is the invariate Harr measure. For detailed study consult Mathai [10].

STATISTICAL PROPERTIES OF GENERALIZED GAMMA DISTRIBUTION INVOLVING PFq
In this section the expression of Mean, Variance, Moment generating function.
Hazard function of generalized gamma distribution are discussed.
Theorem 1: If the random variable x follows the generalized gamma distribution involving
PFq with its respective mean and variance are
p1
m 1, bnp1 Fq , m 1;;k b
Mean =
bm, bn
Fq , m ;; k b
Mean =
m
b
, for small bn (2.1)
k=0 or either =0 or =0 and
b2 m 2, bn m, bn
F 2 m 1, bn,
F
Variance=
p1 q
p1 q
m, bn F *
p1 q
m
=
b2
, for small bn. (2.2)
Here p1 Fq p1Fq , m 2;; k b
q
p1
F
p1
Fq , m 1;; k b
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
ETRASCT' 14 Conference Proceedings
p1 Fq * p1Fq , m ;; k b
Proof : By definition
Mean = Ex xf xdx
0
p1
b1 m 1, bn
F , m 1; ; k b
q
= m, bn F , m ; ; k
p1 q b
Putting k = 0 or either 0 or 0
b1 m 1, bn
m, bn
mn Um 1,2 m , bn b Um,1 m , bn
m
=
b
, for small bn
and Variance = Ex2 Ex2
Ex2 x2 f xdx
m 2, bn F
= p1 q
b2 m, bn F *
p1 q
Putting k=0, 0 or 0
b2 m 2, bn
m, bn
mm 1n2 Um 2,3 m , bn Um,1 m , bn
1 m m d
=
b2
, for small bn
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
ETRASCT' 14 Conference Proceedings
Variance =
m
b2
Theorem 2 : The rth moment about origin and the moment generating function of random varable x following generalized gamma distribution are
m r, bnp1 Fq , m r;; k
' b
r br m, bn F , m ;; k
p1 q b
b
1
t m m, bn1
t b
p1
F , m;; k
q
b t
and M.G.F. =
m, bn F , m ;; k
p1 q b
Theorem 3 : If x follows the generalized gamma distribution involving failure rate function is given by
p Fq its Hazard or
Hx
e bx x m1 x n F ; ; kx
F , m ;; k
(2.3)
q
p
bm m, bn, bx
p1 q b
where
m, n, xis generalized in complete gamma function defined as
x
tm1e t
m, n, x= t n dt
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International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181

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Upadhyaya Lalit Mohan, Dhami H.S. (2001) : MInteartnraitixonaGl JeonurenarlaolfizEnagtiinoeenrisngoRfesMearuchlt&ipTleechnology (IJERT) Hypergeometric Functions, # 1818 IMA Preprints Series, UniversityEoTRfAMSCiTn' 1n4eCsoontfear,ence Proceedings Minneapolis, U.S.A.
