THE I-FUNCTION DISTRIBUTION AND ITS EXTENSIONS
arXiv:1503.01925v1 [math.PR] 6 Mar 2015
P. VELLAISAMY AND K. K. KATARIA Abstract. In this paper we introduce a new probability distribution on (0, ∞), associated with the I-function, namely, the I-function distribution. This distribution generalizes several known distributions with positive support. It is also shown that the distribution of products, quotients and powers of independent I-function variates are I-function variates. Another distribution called the I-function inverse Gaussian distribution is also introduced and studied.
1. Introduction Springer and Thompson [21] showed that the probability density functions (pdf’s) of products and quotients of independent beta, gamma and central Gaussian random variables (rv’s) can be expressed in terms of the Meijer G-function (see [15], p. 21). Mathai and Saxena [13] discussed the distribution of the product of two independent rv’s whose pdf can be expressed as the product of two H-functions; see [6], [15] for definition and more details on the H-function. Carter and Springer [4] introduced the H-function distribution which includes, among others, the gamma, beta, Weilbull, chi-square, exponential and the half-normal distribution as particular cases. Inayat-Hussain [8] generalized the H-function, namely, to the H-function. The generalized Riemann zeta function, the polylogarithmic function of complex order and the exact partition function of the Gaussian free energy model in statistical mechanics are some special cases of the H-function, which are not particular cases of the H-function. Rathie [18] introduced the I-function, which includes the H-function as a special case. The I-function is represented by the following MellinBarnes type contour integral " # Z (ai , Ai , αi )1,p 1 m,n χ(s)z −s ds, (1.1) = I(z) = Ip,q z (bj , Bj , βj )1,q 2πi C
where
Qn
Qm βj αi Γ (1 − a − A s) i i i=1 j=1 Γ (bj + Bj s) Qq . χ(s) = Qp βj α i i=n+1 Γ (ai + Ai s) j=m+1 Γ (1 − bj − Bj s)
(1.2)
An equivalent definition can be obtained on substituting w = −s in (1.1). In the above definition, z 6= 0, m, n, p, q are integers satisfying 0 ≤ m ≤ q and 0 ≤ n ≤ p with αi , Ai > 0 for i = 1, 2, . . . , p and βj , Bj > 0 for j = 1, 2, . . . , q. Also, ai ’s and bj ’s are complex numbers such that no singularity of Γβj (bj + Bj s) coincides with any singularity of Γαi (1 − ai − Ai s). An empty product is to be interpreted as unity. The path of integration C, in the complex s-plane runs from c − i∞ to c + i∞ for some real number c such that the singularity of Γβj (bj + Bj s) lie entirely to the left of the path and the singularity 2010 Mathematics Subject Classification. Primary : 60E05 ; Secondary : 33C60. Key words and phrases. I-function; H-function; Mellin transform. The research of K. K. Kataria was supported by UGC, Govt. of India. 1
of Γαi (1 − ai − Ai s) lie entirely to the right of the path. The conditions for convergence of the integral involved in (1.1) are as follows (see [18]). Let ∇ = Ω =
q X j=1 p
X 1 i=1
∆ =
βj Bj −
m X j=1
2
p X
αi Ai ,
i=1
− Re(ai ) αi −
βj Bj −
q X
q X 1
βj Bj +
j=m+1
2
j=1
n X i=1
− Re(bj ) βj ,
αi Ai −
p X
αi Ai ,
i=n+1
where m, n, ai , Ai , αi and bj , Bj , βj appear in the definition of the I-function. The I-function is analytic if ∇ ≥ 0 and the integral in (1.1) converges absolutely if | arg(z)| < ∆ π2 , where ∆ > 0. Also, if | arg(z)| = ∆ π2 with ∆ ≥ 0, then it converges absolutely under the following conditions: (i) ∇ = 0 and Ω < −1, (ii) |∇| = 6 0 with s = σ + it, where σ, t ∈ R and are such that for |t| → ∞ we have Ω + σ∇ < −1. The H-function follows as a particular case when αi = 1 for i = n + 1, . . . , p and βj = 1 for j = 1, 2, . . . , m in (1.1). Recently, the extension of the I-function to several complex variables (see [11], [17]) and its applications to wireless communication were studied by several authors (see [1], [2], [16]). The effect of Marichev-Saigo-Maeda fractional operators on the I-function has been recently studied by Kataria and Vellaisamy (see [10]). The main purpose of this paper is to introduce a new probability distribution based on the I-function, which we call the I-function distribution. In Section 2, some preliminary results are stated that will be used in subsequent sections. The Mellin and Laplace transforms of the I-function are evaluated in Section 3. In Section 4, the I-function distribution is introduced and some results related to the distribution of products, quotients and powers of independent I-function variates are derived. Also, the I-function inverse Gaussian distribution is introduced and studied in Section 5.
2. Preliminaries The I-function is connected to the generalized hypergeometric function p Fq , the Meijer’s G-function, the generalized Wright function p ψq , the Fox’s H-function and the H-function by the following relationships: # " Qp Γ(ai ) (1 − a , 1, 1) 1,p i 1,p = Qqi=1 Ip,q+1 − z p Fq ((ai )1,p ; (bj )1,q ; z) ; (0, 1, 1) (1 − bj , 1, 1)1,q j=1 Γ(bj ) # # " " (a , 1, 1)1,p m,n (ai )1,p ; = Gm,n Ip,q z i p,q z (b ) (bj , 1, 1)1,q j 1,q # # " " (ai , Ai )1,p (1 − ai , Ai , 1)1,p 1,p ; = p ψq z Ip,q+1 − z (bj , Bj )1,q (0, 1, 1) (1 − bj , Bj , 1)1,q " # " # (a , A ) (a , A , 1) i i 1,p i i 1,p m,n m,n z ; z = Hp,q Ip,q (bj , Bj )1,q (bj , Bj , 1)1,q 2
and # # " " m,n (a , A , α ) (ai , Ai )n+1,p (a , A , α ) (ai , Ai , 1)n+1,p m,n . = H p,q z i i i 1,n Ip,q z i i i 1,n (bj , Bj )1,m (bj , Bj , βj )m+1,q (bj , Bj , 1)1,m (bj , Bj , βj )m+1,q
The following identities will be used later: " # " # 1 b 1 (b − a + 1, 1, 1) 1,0 1,1 z = z B e−z B ; I1,1 I0,1 z = Γ(a)z b (1 + z)−a ; (b, B, 1) (b, 1, 1) B # " (b + 1, 1, 1) 0,1 = z b , |z| > 1; I1,1 z (b, 1, 1)
and
Γ(a +
1,0 1)δ a+b I1,1
# " z (a + b + 1, 1, 1) = z b (δ − z)a , |z| < δ. (b, 1, 1) δ
Also, from the definition of the I-function, the following properties are immediate. # # " " 1 (1 − b , B , β ) (a , A , α ) j j j 1,q n,m m,n i i i 1,p ; = Iq,p z Ip,q (1 − ai , Ai , αi )1,p z (bj , Bj , βj )1,q # " " # −1 (a , σ A , α ) 1 (a , A , α ) i i i 1,p m,n = Ip,q z σ i i i 1,p I m,n z , σ > 0; (bj , σ −1 Bj , βj )1,q (bj , Bj , βj )1,q σ p,q " # " # (a + σA , A , α ) (a , A , α ) i i i 1,p m,n m,n z i . z i i i 1,p = Ip,q z σ Ip,q (bj + σBj , Bj , βj )1,q (bj , Bj , βj )1,q
(2.1)
(2.2)
(2.3)
(2.4) (2.5) (2.6)
Note that whether a product of two I-functions is another I-function is still an open problem. 3. The Mellin and Laplace transform of the I-Function In this section, we evaluate the Mellin and Laplace transform of the I-function. The Mellin and Laplace transform of a continuous positive rv X with pdf fX is defined by Z ∞ s−1 MfX (s) = E[X ] = xs−1 fX (x) dx (3.1) 0
and
−rX
LfX (r) = E[e
]=
Z
∞
e−rx fX (x) dx,
(3.2)
0
respectively. Under some suitable restrictions [22] on MfX (s), there exists an inversion integral Z c+i∞ 1 fX (x) = x−s MfX (s) ds. (3.3) 2πi c−i∞ The following results are due to Epstein [5]. (i) The Mellin transform of the positive rv Y = aX, a > 0 is MfY (s) = as−1 MfX (s).
(3.4)
MfY (s) = MfX (rs − r + 1).
(3.5)
(ii) The Mellin transform of the positive rv Y = X r is 3
(iii) Let X1 , X2 be two positive continuous independent rv’s. The Mellin transform of the rv Y = X1 /X2 is MfY (s) = MfX1 (s)MfX2 (2 − s). (3.6) (iv) Let X1 , X2 , . . . ,Q Xn be n positive continuous independent rv’s. Then the Mellin transform of the rv Y = nj=1 Xj is MfY (s) =
n Y j=1
MfXj (s).
(3.7)
3.1. The Mellin transform. Assuming the convergence of the integral involved in definition of the I-function, for σ > 0 and δ 6= 0, we have # " (a , A , α ) m,n (3.8) δxσ i i i 1,p I(δxσ ) = Ip,q (bj , Bj , βj )1,q Z 1 = χ(w)δ −w x−σw dw. 2πi C
Substituting s = σw in the above integral and using (3.3), the Mellin transform of the I-function (3.8) is χ(σ −1 s) σ MI(δx ) (s) = . (3.9) σδ σ−1 s The above result can be extended to obtain the Mellin transform of product of two Ifunctions. We state the following useful result in the form of a lemma. Lemma 3.1. The Mellin transform, MI(δxσ )I(ηxµ ) (s), of the product of # " # " (c , C , γ ) (a , A , α ) k,l m,n ηxµ i i i 1,u and I(ηxµ ) = Iu,v δxσ i i i 1,p I(δxσ ) = Ip,q (dj , Dj , ρj )1,v (bj , Bj , βj )1,q
for σ > 0, µ ≥ 0 and δ, η 6= 0 is equal to # " −1 −1 η 1 (ci , Ci , γi )1,l (1 − bj − σ Bj s, µσ Bj , βj )1,q (ci , Ci , γi )l+1,u . I n+k,m+l σδ σ−1 s q+u,p+v δ σ−1 µ (dj , Dj , ρj )1,k (1 − ai − σ −1 Ai s, µσ −1Ai , αi )1,p (dj , Dj , ρj )k+1,v Proof. From (3.1), we have # # " " Z ∞ (c , C , γ ) (a , A , α ) k,l m,n ηxµ i i i 1,u dx MI(δxσ )I(ηxµ ) (s) = xs−1 Ip,q δxσ i i i 1,p Iu,v (dj , Dj , ρj )1,v (b , B , β ) j j j 1,q 0 # " Z ∞ Z 1 s−1 m,n σ (ai , Ai , αi )1,p = x Ip,q δx χ∗ (w)(ηxµ )−w dw dx, (b , B , β ) 2πi j j j 1,q 0 C
where
Qk Ql ρj γi j=1 Γ (dj + Dj w) i=1 Γ (1 − ci − Ci w) Q . χ (w) = Qu v ρj γi j=k+1 Γ (1 − dj − Dj w) i=l+1 Γ (ci + Ci w) We assume the conditions for absolute convergence of integral involved in above equation holds. By interchanging the order of integrals, we get # " Z ∞ Z 1 (a , A , α ) i i i 1,p m,n MI(δxσ )I(ηxµ ) (s) = dx dw δxσ χ∗ (w)η −w xs−µw−1 Ip,q (bj , Bj , βj )1,q 2πi C 0 ∗
4
Z 1 = χ∗ (w)η −w MI(δxσ ) (s − µw) dw. 2πi C Z −w 1 1 ∗ −1 −1 −σ−1 µ χ (w)χ(σ s − σ µw) ηδ dw. = σδ σ−1 s 2πi C
The proof is complete by using definition of the I-function (1.1).
3.2. The Laplace transform. For simplicity, we use an equivalent definition of the Ifunction to determine its Laplace transform. Moreover, the conditions for absolute convergence of integral involved in the definition of I(δxσ ) is assumed. From (3.2), we have Z ∞ LI(δxσ ) (r) = e−rx I(δxσ ) dx 0 Z ∞ −rx Z e = χ(−s)(δxσ )s ds dx 2πi 0 C Z ∞ Z 1 s χ(−s)δ = xσs e−rx dx ds 2πi C 0 s Z δ 1 1 Γ(1 + σs)χ(−s) σ ds = r 2πi C r " # 1 m,n+1 δ (0, σ, 1) (ai , Ai , αi )1,p = I r p+1,q r σ (bj , Bj , βj )1,q # " r σ−1 m,n+1 δ (σ, σ, 1) (ai + Ai , Ai , αi )1,p (using (2.6)) I = δ p+1,q r σ (bj + Bj , Bj , βj )1,q # " r σ−1 n+1,m r σ (1 − bj − Bj , Bj , βj )1,q = , (3.10) I (1 − σ, σ, 1) (1 − ai − Ai , Ai , αi )1,p δ q,p+1 δ where the last step follows from (2.4).
4. The I-Function Distribution Here, we introduce a new probability distribution associated with the I-function. Let X > 0 be a rv with pdf # " (a , A , α ) m,n δxσ i i i 1,p , x > 0, k0 Ip,q (bj , Bj , βj )1,q (4.1) fX (x) = 0 otherwise,
where σ > 0, δ 6= 0 and (ai , Ai , αi )1,p , (bj , Bj , βj )1,q , m, n must conform to those restrictions in R ∞the definition of the I-function (1.1), such that fX (x) ≥ 0. Also, k0 is such that fX (x) dx = 1. 0
(a) Moments. The r-th moment for the I-function distribution can be obtained using the Mellin integral transform as follows. µ′r = MfX (r + 1) = k0 MI(δxσ ) (r + 1) 5
=
k0 χ(σ −1 + σ −1 r) , σδ (r+1)σ−1
using (3.9) and χ(.) is defined in (1.2). The normalizing constant of the I-function distribution is obtained from µ′0 = 1 as σδ σ . χ(σ −1 ) −1
k0 =
(b) The characteristic function. The characteristic function of the I-function distribution is given by φX (t) = k0 LI(δxσ ) (−it) =
k0 (−it) δ
σ−1
n+1,m Iq,p+1
"
# (−it) (1 − bj − Bj , Bj , βj )1,q , (1 − σ, σ, 1) (1 − ai − Ai , Ai , αi )1,p δ σ
where the last step follows from (3.10). Hence, the moment generating function is MX (t) = φX (−it)
# " k0 (−t)σ−1 n+1,m (−t)σ (1 − bj − Bj , Bj , βj )1,q . = Iq,p+1 (1 − σ, 1, 1) (1 − ai − Ai , Ai , αi )1,p δ δ
(c) The cumulative distribution function (cdf ). The Mellin transform of the cdf FX (x) of a rv X is connected to its pdf fX (x) by the following relationship (see [20], p. 99). M1−FX (x) (s) = s−1 MfX (s + 1).
(4.2)
Therefore the cdf of the I-function distribution is given by # " −1 −1 k0 m+1,n −1 (a + σ A , σ A , α ) (1, 1, 1) i i i 1,p . FX (x) = 1 − σ−1 Ip+1,q+1 δ σ x i (0, 1, 1) (bj + σ −1 Bj , σ −1 Bj , βj )1,q σδ
And hence, the survival function is " # −1 −1 k0 m+1,n −1 (a + σ Ai , σ Ai , αi )1,p (1, 1, 1) F X (x) = σ−1 Ip+1,q+1 δ σ x i . −1 (0, 1, 1) (b + σ Bj , σ −1 Bj , βj )1,q σδ j The hazard rate function λX (x) is given by λX (x) =
fX (x) F X (x) σδ
= m+1,n Ip+1,q+1
"
δ
σ−1
σ−1
m,n Ip,q
# (a , A , α ) δxσ i i i 1,p (bj , Bj , βj )1,q
"
#. (ai + σ −1 Ai , σ −1 Ai , αi )1,p (1, 1, 1) x (0, 1, 1) (bj + σ −1 Bj , σ −1 Bj , βj )1,q 6
(d) Order statistics. Let X(1) ≤ X(2) ≤ . . . ≤ X(N ) denotes the order statistics of a random sample X1 , X2 , . . . , XN of size N drawn from the I-function distribution. The pdf of the j-th order statistics is given by fX(j) (x) =
N −j N! fX (x) {FX (x)}j−1 F X (x) . (j − 1)!(N − j)!
Therefore, the pdf of the minimum and the maximum order statistics is given by # " Nk0N (a , A , α ) I m,n δxσ i i i 1,p fX(1) (x) = (bj , Bj , βj )1,q σ N −1 δ (N −1)σ−1 p,q #)N −1 ( " −1 −1 −1 (a + σ A , σ A , α ) (1, 1, 1) m+1,n i i i i 1,p Ip+1,q+1 δ σ x (0, 1, 1) (bj + σ −1 Bj , σ −1 Bj , βj )1,q and
#( (ai , Ai , αi )1,p k0 1 − σ−1 fX(N) (x) = δx (bj , Bj , βj )1,q σδ #)N −1 " −1 −1 −1 (1, 1, 1) (a + σ Ai , σ Ai , αi )1,p m+1,n Ip+1,q+1 δ σ x i . (0, 1, 1) (bj + σ −1 Bj , σ −1 Bj , βj )1,q m,n Nk0 Ip,q
"
σ
4.1. Some special cases. The density of the H-function distribution (see [4]), # " (a , A ) i i 1,p δ m,n , x > 0, Hp,q δx χ(1) (bj , Bj )1,q fX (x) = 0 otherwise,
(4.3)
(ii) the exponential distribution (put θ = 1 in (4.4)) with parameter λ > 0. Then " # 1 x 1 1,0 x fX (x) = e− λ = I0,1 , x > 0. λ λ λ (0, 1, 1)
(4.5)
follows as special case of the I-function distribution (4.1) when σ = 1, αi ’s and βj ’s for all i and j equals unity. Hence, the other standard non-negative distributions are particular cases. We next give the densities of the well-known distributions in terms of the I-function. Let X be a rv which follows (i) the gamma distribution with parameters θ, λ > 0. Then " # x 1 1 1,0 x θ−1 − λ , x > 0. (4.4) x e = I fX (x) = θ λ Γ(θ) λΓ(θ) 0,1 λ (θ − 1, 1, 1)
(iii) the chi-square distribution (put θ = denotes the degree of freedom. Then
ν 2
and λ = 2 in (4.4)) with parameter ν ∈ N
" x ν 1 x 1 1,0 −1 − 2 2 e = x I fX (x) = ν 2Γ ν2 0,1 2 2 2 Γ ν2 7
ν 2
− 1, 1, 1
#
, x > 0.
(iv) the Weibull distribution with parameters θ, λ > 0. Then " # 1 1,0 x θ θ−1 − xθ , x > 0. fX (x) = x e λ = 1 I0,1 1 1 1 1 − , , 1 λ θ θ λ λ θ θ
(4.6)
(v) the Rayleigh distribution (put θ = 2 and λ = 2ν 2 in (4.6)) with parameter ν > 0. Then # " 1 1,0 x 1 − x22 fX (x) = 2 xe 2ν = √ I0,1 √ 1 1 , x > 0. ν ν 2 ν 2 2, 2, 1
(vi) the Maxwell distribution with parameter λ > 0. Then # " x2 4 2 x 1,0 2 −λ , x > 0. fX (x) = 3 √ x e 2 = √ I0,1 λ π λ π λ 1, 21 , 1
(vii) the half-normal distribution with parameter λ > 0. Then # " √ 2 − x22 x 1 1,0 fX (x) = √ e 2λ = √ I0,1 √ 1 , x > 0. λ π λ 2π λ 2 0, 2 , 1 (viii) the half-Cauchy distribution with parameter λ > 0. Then " # 1 1 1,1 x 0, 21 , 1 2λ = I , x > 0. fX (x) = π (λ2 + x2 ) λπ 1,1 λ 0, 21 , 1
(4.7)
(4.8)
(ix) the beta distribution of first kind with parameters θ, λ > 0. Then # " Γ(θ + λ) Γ(θ + λ) θ−1 (θ + λ − 1, 1, 1) , 0 < x < 1. x (1 − x)λ−1 = I 1,0 x fX (x) = Γ(θ)Γ(λ) Γ(θ) 1,1 (θ − 1, 1, 1)
(x) the beta distribution of second kind with parameters θ, λ > 0. Then # " θ −(θ+λ) Γ(θ + λ) λ λx λ 1,1 λx (−λ, 1, 1) θ−1 fX (x) = , x > 0. 1+ I x = Γ(θ)Γ(λ) θ θ θΓ(θ)Γ(λ) 1,1 θ (θ − 1, 1, 1)
(xi) the power function distribution with parameter θ > 0. Then # " (θ, 1, 1) 1,0 , 0 < x < 1. fX (x) = θxθ−1 = θI1,1 x (θ − 1, 1, 1) (xii) the uniform distribution on (0, 1) (put θ = 1 in (4.9)). Then # " (1, 1, 1) 1,0 , 0 < x < 1. x fX (x) = 1 = I1,1 (0, 1, 1)
(xiii) the Pareto distribution with parameter λ > 0. Then " # (−λ, 1, 1) 0,1 x , x > 1. fX (x) = λx−(λ+1) = λI1,1 (−λ − 1, 1, 1)
(xiv) the half-student distribution with parameter ν > 0. Then " 1−ν 1 # ν+1 2 − 2 2Γ ν+1 x 1 x 1,1 2 2 ,1 2 , 1 , x > 0. √ 1 + = I fX (x) = √ √ 1,1 ν ν ν ν 0, 2 , 1 νπΓ 2 νπΓ 2 8
(4.9)
(4.10)
(xv) the F distribution with parameters θ, λ > 0. Then " θ λ # λ 2λ2 (θ+λ) θ Γ θ+λ θ θ 1,1 θx − 2 , 1, 1 2 x 2 −1 (λ + θx)− 2 = λ I1,1 , x > 0. fX (x) = λ θ θ λ θ2 − 1, 1, 1 Γ 2 Γ 2 λΓ 2 Γ 2 (xvi) the general hypergeometric distribution [14]. Then c da d Γ(α)Γ β − dc c−1 d fX (x) = x F α; β; −ax 1 1 Γ dc Γ(β)Γ α − dc " 1 1 − α + c−1 , 1 , 1 a d Γ β − dc 1 1,1 d d I = ad x c−1 1 , d, 1 Γ dc Γ α − dc 1,2 d
1−β+
c−1 1 , d, 1 d
#
, x > 0.
The above relationships can easily be obtained using identities (2.1)-(2.3) and properties of the I-function (2.4)-(2.6). 4.2. Properties of the I-function distribution. Next we show that the class of Ifunction variates is closed under multiplication and quotients of independent I-function variates. Moreover, this class is also closed under positive scalar multiplication and rational power of the I-function variate. Theorem 4.1. Let X1 , X2 , . . . , XN be N independent I-function rv’s with pdf’s fX1 (x1 ), fX2 (x2 ), . . . , fXN (xN ), respectively, where " # σ−1 σl δl l σl (ali , Ali , αli )1,pl ml ,nl I δl xl , xl > 0, (4.1) fXl (xl ) = (blj , Blj , βlj )1,ql χl (σl−1 ) pl ,ql and
Q l β lj (b Γαli (1 − ali − Ali s) m lj + Blj s) j=1 Γ Q Q χl (s) = pl , q l βlj (1 − b − B s) αli (a + A s) lj lj li li j=ml +1 Γ i=nl +1 Γ Q for l = 1, 2, . . . , N. Then, the pdf of the rv Y = N l=1 Xl is given by # " (a , A , α ) m,n fY (y) = kIp,q δy i i i 1,p , y > 0, (bj , Bj , βj )1,q Qnl
i=1
σ −1
δl l l=1 χl (σ−1 ) , l
PN PN PN PN m = where k = l=1 ml , n = l=1 nl , p = l=1 pl , q = l=1 ql , δ = QN σl−1 and the sequence of parameters are as follows: l=1 δl QN
(ai , Ai , αi )1,p = (a1i , σl−1 A1i , α1i )1,n1 (a2i , σl−1 A2i , α2i )1,n2 . . . (aN i , σl−1 AN i , αN i )1,nN (a1i , σl−1 A1i , α1i )n1 +1,p1 (a2i , σl−1 A2i , α2i )n2 +1,p2 . . . (aN i , σl−1 AN i , αN i )nN +1,pN , (bj , Bj , βj )1,q = (b1j , σl−1 B1j , β1j )1,m1 (b2j , σl−1 B2j , β2j )1,m2 . . . (bN j , σl−1 BN j , βN j )1,mN (b1j , σl−1 B1j , β1j )m1 +1,q1 (b2j , σl−1 B2j , β2j )m2 +1,q2 . . . (bN j , σl−1 BN j , βN j )mN +1,qN .
Proof. From (3.7), we have for y > 0 MfY (s) =
N Y l=1
σ N Y δl l χl (σl−1 s) −1
MfXl (s) = 9
σl−1 s χl (σl−1 ) l=1 δl
.
Now using the inverse Mellin transform (3.3), the pdf is given by !−s ! Z c+i∞ Y σ−1 N N N Y Y −1 δl l 1 σ δl l ds, χl (σl−1 s) y fY (y) = −1 2πi χ (σ ) l c−i∞ l l=1 l=1 l=1 which on using (1.1) gives the result.
Example 4.1. Let X1 , X2 , . . . , Xn be n independent rv’s such that Xi follows half-normal distribution for i = 1, 2, . . . , m and beta distribution for i = m + 1, . . . , n, with pdf’s √ 2 2 − 2σxi2 √ e i , xi > 0, i = 1, 2, . . . , m, fXi (xi ) = σi π Γ(ai + bi ) ai −1 and fXi (xi ) = x (1 − xi )bi −1 , 0 < xi < 1, i = m + 1, . . . , n, Γ(ai )Γ(bi ) i where the parameters σi , ai , bi > 0. The corresponding Mellin transforms (3.1) are given by −s 1 1 Γ 2s , i = 1, 2, . . . , m, √2πσi √2σi MfXi (s) = Γ(ai +bi )Γ(ai −1+s) , i = m + 1, . . . , n. Γ(ai )Γ(ai +bi −1+s) Q Let Y = ni=1 Xi . Using (3.7), its Mellin transform is n m √ Y ( 2σi )s Y Γ(ai + bi )Γ(ai − 1 + s) m s √ . (4.2) MfY (s) = Γ 2 i=1 2πσi i=m+1 Γ(ai )Γ(ai + bi − 1 + s) Using (3.3) and (4.2), the pdf of rv Y is c+i∞
m Y
n s Y
!−s
1 Γ(ai − 1 + s) √ ds (4.3) y 2 i=m+1 Γ(ai + bi − 1 + s) 2σi c−i∞ i=1 " # m 2− 2 y (ai + bi − 1, 1,1)m+1,n n−m+1,0 , y > 0, = kIn−m,n−m+1 Qm (ai − 1, 1, 1)m+1,n 0, 21 , m i=1 σi
fY (y) =
k 2πi
Z
Γm
Q Qn Γ(ai +bi ) √1 where k = m i=1 2πσi i=m+1 Γ(ai ) . Note that the integral (4.3) can also be represented in terms of the Fox’s H-function as # " Qn Γ(ai +bi ) m 2− 2 y (ai + bi − 1, 1)m+1,n i=m+1 Γ(ai ) n,0 , y > 0. fY (y) = Hn−m,n Qm m Q (ai − 1, 1)m+1,n (0, 21 )1,m (2π) 2 m i=1 σi i=1 σi
Example 4.2. Let X1 , X2 be two independent rv’s such that X1 is uniformly distributed (4.10) on (0, 1) and X2 follows gamma distribution (4.4) with shape parameter θ = 2. Using Theorem 4.1, the pdf of rv Y = X1 X2 is # " 1 1,0 y fY (y) = I0,1 , y > 0, λ λ (0, 1, 1) which is an exponential distribution with parameter λ (see [9], p. 306). 10
Theorem 4.2. Let X be an I-function rv with pdf (4.1). (a) The pdf of the rv Y = aX, a > 0, is given by # " −1 −1 σ −1 δσ δ y (a , σ A , α ) i i 1,p i fY (y) = , y > 0. I m,n aχ(σ −1 ) p,q a (bj , σ −1 Bj , βj )1,q
(b) The pdf of the rv Y = X r for r 6= 0 rational is given by # " −1 −1 −1 −1 δ rσ −1 (a − rσ A + σ A , rσ A , α ) i i i i i 1,p fY (y) = , y > 0, I m,n δ rσ y (bj − rσ −1 Bj + σ −1 Bj , rσ −1Bj , βj )1,q χ(σ −1 ) p,q
(4.4)
(4.5)
when r > 0 and
" # −1 −1 −1 −1 δ rσ rσ−1 (1 − bj + rσ Bj − σ Bj , −rσ Bj , βj )1,q n,m δ y , y > 0, (4.6) I fY (y) = (1 − ai + rσ −1 Ai − σ −1 Ai , −rσ −1 Ai , αi )1,p χ(σ −1 ) q,p
when r < 0.
Proof. (a) From (3.4), we have as−1 δ σ χ(σ −1 s) . δ σ−1 s χ(σ −1 ) −1
MfY (s) = as−1 MfX (s) = Using (3.3), the pdf of Y is δσ 1 fY (y) = aχ(σ −1 ) 2πi −1
Z
c+i∞
δσ y a −1
χ(σ −1 s)
c−i∞
!−s
ds,
y > 0,
which gives (4.4) on using (1.1). (b) From (3.5), we have δ rσ χ(σ −1 rs − σ −1 r + σ −1 ) MfY (s) = MfX (rs − r + 1) = . δ rσ−1 s χ(σ −1 ) −1
Therefore, δ rσ 1 fY (y) = −1 χ(σ ) 2πi −1
Z
c+i∞
c−i∞
−1 −s χ(σ −1 rs − σ −1 r + σ −1 ) yδ rσ ds.
(4.7)
When r > 0, the lhs of (4.7) on using (1.1) gives (4.5). When r < 0, put t = −r so that for t > 0, Z c+i∞ −1 −s 1 δ −tσ −1 −1 −1 −tσ−1 χ(−σ ts + σ t + σ ) yδ ds, y > 0, fY (y) = χ(σ −1 ) 2πi c−i∞ which on using (1.1) gives (4.6).
Example 4.3. Let Y be a rv defined in Example 4.1. Consider W = 4Y and Z = Y −1 . Using (4.4) the pdf of rv W is # " m k n−m+1,0 2− 2 w (ai + bi − 1, 1,1)m+1,n fW (w) = In−m,n−m+1 Qm , w > 0, 0, 12 , m (ai − 1, 1, 1)m+1,n 4 4 i=1 σi 11
Q Qn Γ(ai +bi ) √1 where k = m i=1 2πσi i=m+1 Γ(ai ) . Similarly by using (4.6) the pdf of rv Z is " m # 1 Y √ 0, 2 , m (−ai , 1, 1)m+1,n 0,n−m+1 z fZ (z) = k ∗ In−m+1,n−m 2σi , z > 0, (−ai − bi , 1, 1)m+1,n i=1 q Q Q 2 i +bi ) where k ∗ = m σ ni=m+1 Γ(a . i=1 π i Γ(ai )
Example 4.4. Let X be an exponential rv with pdf given by (4.5). Consider Y = θX and 1 Z = X θ where θ > 0. Using (4.4) the pdf of rv Y is # " 1 1,0 y , y > 0, fY (y) = I0,1 β β (0, 1, 1) where β = λθ. Now by using (4.5) the pdf " 1 1,0 z fZ (z) = 1 I0,1 1 λθ λθ
of rv Z is # 1 − 1 , 1 , 1 , z > 0. θ θ
Thus, Y is exponentially distributed with parameter β, whereas Z follows the Weibull distribution (4.6) with parameters λ, θ (see [12]). Theorem 4.3. Let X1 and X2 be two independent I-function rv’s with pdf’s given by (4.1) X1 is given by for l = 1, 2. Then, the pdf of the rv Y = X 2 " # (a , A , α ) m,n fY (y) = kIp,q δy i i i 1,p , y > 0, (bj , Bj , βj )1,q σ −1 −σ −1
δ1 1 δ2 2 , χ1 (σ1−1 )χ2 (σ2−1 )
σ−1 −σ2−1
where k = m = m1 + n2 , n = n1 + m2 , p = p1 + q2 , q = q1 + p2 , δ = δ1 1 δ2 and the sequence of parameters are
(ai , Ai , αi )1,p = (a1i , σ1−1 A1i , α1i )1,n1 (1−b2j −2σ2−1 B2j , σ2−1 B2j , β2j )1,q2 (a1i , A1i , σ1−1 α1i )n1 +1,p1 ,
(bj , Bj , βj )1,q = (b1j , σ1−1 B1j , β1j )1,m1 (1−a2i −2σ2−1 A2i , σ2−1 A2i , α2i )1,p2 (b1j , σ1−1 B1j , β1j )m1 +1,q1 . Proof. From (3.6), we have for y > 0, fY (y) = M−1 MfX1 (s)MfX2 (2 − s) Z c+i∞ σ−1 −σ−1 −1 −s δ1 1 δ2 2 1 σ1 −σ2−1 −1 −1 −1 = ds, χ (σ s)χ (2σ − σ s) yδ δ 1 2 1 2 2 1 2 χ1 (σ1−1 )χ2 (σ2−1 ) 2πi c−i∞
which on using the definition of the I-function (1.1) completes the proof.
Example 4.5. Let X1 , X2 be two independent half-normal rv’s (4.7) with pdf’s # " x 1 1,0 √i 1 , xi > 0, fXi (xi ) = √ I0,1 λi 2π λi 2 0, 2 , 1
where the parameters λi > 0 for i = 1, 2. Using Theorem 4.3, the rv Y = X1 /X2 follows Cauchy distribution (4.8) with pdf " # 1 1,1 y 0, 21 , 1 fY (y) = , y > 0, I λπ 1,1 λ 0, 21 , 1 12
where the parameter λ = λ1 /λ2 (see [12]). 4.3. The distribution based on the product of two I-functions. Let X be a rv with pdf # # " " (c , C , γ ) (a , A , α ) k,l m,n ηxµ i i i 1,u , x > 0, δxσ i i i 1,p Iu,v k1 Ip,q (dj , Dj , ρj )1,v (bj , Bj , βj )1,q (4.8) fX (x) = 0 otherwise,
where σ > 0, µ ≥ 0 and δ, η 6= 0. Also, (ai , Ai , αi )1,p , (bj , Bj , βj )1,q , (ci , Ci , γi )1,u , (dj , Dj , ρj )1,v , m, n, k, l must conform to those restrictions in the definition of the I-function (1.1), such that fX (x) ≥ 0. Using Lemma 3.1, the normalizing constant k1 is # " −1 −1 −1 σ η (c , C , γ ) (1 − b − σ B , µσ B , β ) (c , C , γ ) n+k,m+l j j j j 1,q i i i l+1,u i i i 1,l k1−1 = σ−1 Iq+u,p+v . δ δ σ−1 µ (dj , Dj , ρj )1,k (1 − ai − σ −1 Ai , µσ −1 Ai , αi )1,p (dj , Dj , ρj )k+1,v
Special cases. (i) The generalized gamma distribution associated with Bessel function (see [19]) has density fX (x) =
aα Γ(α)e −λ a
= ae
λ a
xα−1 e−ax 0 F1 (−; α; λx) ,
1,0 I0,1
x > 0,
# # " , ax I 1,0 − λx (0, 1, 1) (1 − α, 1, 1) (α − 1, 1, 1) 0,2
"
(4.9)
where α, a > 0 and λ 6= 0. (ii) The non-central chi-square distribution put α = ν2 , a = 12 and λ = β4 in (4.9) is (x+β) ν 1 ν βx −1 − fX (x) = , x > 0, x 2 e 2 0 F1 −; ; ν 2 4 2 2 Γ( ν2 ) " # " # − β2 e βx 1,0 x 1,0 = I0,1 I0,2 − , (4.10) ν ν ( − 1, 1, 1) 2 2 2 4 (0, 1, 1) (1 − 2 , 1, 1)
where ν ∈ N is the degree of freedom and β > 0. (iii) The generalized gamma distribution associated with confluent hypergeometric function (see [7]) has density aβ (a + 1)α−β α−1 −(a+1)x x e x > 0, fX (x) = 1 F1 (β; α; x) , Γ(α) " # # " (1 − β, 1, 1) aβ (a + 1)1−β 1,0 1,1 = I0,1 (a + 1)x , I − x (0, 1, 1) (1 − α, 1, 1) (α − 1, 1, 1) 1,2 Γ(β)
(4.11)
where α, β, a > 0. Remark 4.1. For k = u = v = 1, l = µ = 0, |η| < 1 and (c1 , C1 , γ1) = (1, 1, 1), (d1, D1 , ρ1 ) = (0, 1, 1) the pdf (4.8) reduces to (4.1). 13
5. The I-Function inverse Gaussian Distribution Consider a rv X > 0 with pdf # " (a , A , α ) −1 m,n δxσ i i i 1,p , x > 0, k2 xα−1 e−ax−bx Ip,q (bj , Bj , βj )1,q fX (x) = 0 otherwise,
(5.1)
with the following restriction on parameters:
a > 0, b ≥ 0 if α > 0, a ≥ 0, b > 0 if α < 0, a > 0, b > 0 if α = 0.
Further, σ ≥ 0, δ 6= 0 and (ai , Ai , αi )1,p , (bj , Bj , βj )1,q , m, n must conform to those restrictions in the definition of the I-function (1.1), such that fX (x) ≥ 0. In order to obtain the normalizing constant k2 of the I-function inverse Gaussian (I-FIG) distribution, we first determine its Mellin transform. Theorem 5.1. The Mellin transform of the I-FIG distribution is MfX (s) ( " # ∞ X δ (2 − α − s, σ, 1) (a , A , α ) (2 + l − α − s, σ, 1) m+1,n+1 i i i 1,p a−α−s+1 Ip+2,q+1 = k2 σ (2 − α − s, σ, 1) (bj , Bj , βj )1,q a l=0 #) " (ab)l (2 − α − s, σ, 1) (a , A , α ) m+1,n+1 i i i 1,p . − bα+s−1 Ip+1,q+2 bσ δ (2 − α − s, σ, 1) (bj , Bj , βj )1,q (1 − l − α − s, σ, 1) l!
Proof. Using the definition of Mellin transform and assuming the conditions for absolute convergence of integral involved, we have # " Z ∞ σ (ai , Ai , αi )1,p α+s−2 −ax−bx−1 m,n dx MfX (s) = k2 x e Ip,q δx (bj , Bj , βj )1,q 0 Z ∞ Z 1 α+s−2 −ax−bx−1 σ −w = k2 x e χ(w)(δx ) dw dx 2πi C 0 Z Z ∞ k2 α+s−σw−2 −ax−bx−1 = x e dx χ(w)δ −w dw. (5.2) 2πi C 0 Now r !α+s−σw−1 Z ∞ √ b −1 (5.3) xα+s−σw−2 e−ax−bx dx = 2 Kα+s−σw−1 (2 ab), a 0
where Kν (x) is the modified Bessel function of the second kind, also called Macdonald function, defined by π (I−ν (x) − Iν (x)) Kν (x) = , (5.4) 2 sin νπ where Iν (x) is the modified Bessel function of the first kind with the following series representation: ∞ x 2l+ν X 1 . (5.5) Iν (x) = Γ(l + ν + 1)l! 2 l=0 14
Using (5.3)-(5.5) and gamma reflection principle in (5.2), we have ( −w Z X ∞ δ k2 −α−s+1 Γ(α + s − σw − 1)Γ(2 − α − s + σw)χ(w) a MfX (s) = 2πi C l=0 Γ(l − α − s + σw + 2) aσ (ab)l α+s−1 Γ(α + s − σw − 1)Γ(2 − α − s + σw)χ(w) σ −w −b (b δ) dw. Γ(l + α + s − σw) l!
Finally, by using the definition of the I-function (1.1), the proof is complete.
Corollary 5.1. The r-th moment for the I-FIG distribution is # " ( ∞ X δ (1 − α − r, σ, 1) (ai , Ai , αi )1,p (1 + l − α − r, σ, 1) m+1,n+1 a−α−r Ip+2,q+1 µ′r = k2 aσ (1 − α − r, σ, 1) (bj , Bj , βj )1,q l=0 #) " (ab)l σ (1 − α − r, σ, 1) (ai , Ai , αi )1,p α+r m+1,n+1 . − b Ip+1,q+2 b δ (1 − α − r, σ, 1) (bj , Bj , βj )1,q (−l − α − r, σ, 1) l! Hence, the normalizing constant of the I-FIG distribution is given by # " ( ∞ X δ (1 − α, σ, 1) (a , A , α ) (1 + l − α, σ, 1) m+1,n+1 i i i 1,p a−α Ip+2,q+1 k2−1 = σ (1 − α, σ, 1) (bj , Bj , βj )1,q a l=0 #) " (ab)l (1 − α, σ, 1) (a , A , α ) m+1,n+1 i i i 1,p bσ δ − bα Ip+1,q+2 . (1 − α, σ, 1) (bj , Bj , βj )1,q (−l − α, σ, 1) l!
Corollary 5.2. The Laplace transform of the I-FIG distribution is " # ( ∞ X δ 1 (1 − α, σ, 1) (ai , Ai , αi )1,p (1 + l − α, σ, 1) m+1,n+1 LfX (r) = k2 I (a + r)α p+2,q+1 (a + r)σ (1 − α, σ, 1) (bj , Bj , βj )1,q l=0 #) " ((a + r)b)l (1 − α, σ, 1) (a , A , α ) i i i 1,p σ α m+1,n+1 . − b Ip+1,q+2 b δ (1 − α, σ, 1) (bj , Bj , βj )1,q (−l − α, σ, 1) l!
Special cases. The generalized inverse Gaussian distribution studied by Barndorff-Nielsen [3]. α
1 β2 −1 fX (x) = √ xα−1 e− 2 (βx+γx ) α 2 2γ Kα ( βγ)
# " α γ −1 β β2 (1, 1, 1) 1,0 = √ , xα−1 e− 2 x− 2 x I1,1 δ α (0, 1, 1) 2γ 2 Kα ( βγ)
x > 0,
(5.6)
where |δ| < 1, β, γ > 0, α ∈ R. Therefore the inverse Gaussian, inverse gamma and hyperbolic distributions follow as particular cases. (i) The inverse Gaussian distribution put α = − 12 , β = µλ2 and γ = λ in (5.6) . r 2 λ − λ(x−µ) 2µ2 x , fX (x) = e 2πx3 " # r λ λµ − 1 −1 − 2µλ2 x− λ2 x−1 1,0 (1, 1, 1) , x > 0, I1,1 δ e x 2 e = (0, 1, 1) 2π 15
where |δ| < 1 and λ, µ > 0. Note that K− 1 (x) = 2 (ii) The inverse Gamma distribution. λθ −θ−1 −λx−1 x e , fX (x) = Γ(θ)
pπ 2
1
e−x x− 2 .
" # λθ −θ−1 −λx−1 1,0 (1, 1, 1) , = x e I1,1 δ (0, 1, 1) Γ(θ)
x > 0,
where |δ| < 1 and θ, λ > 0.
Remark 5.1. When b = 0, the I-FIG distribution in (5.1) belongs to the class of distributions defined by the product of two I-functions (4.8) as follows. " # " # (a , A , α ) 1,0 ax fX (x) = k∗ I0,1 I m,n δxσ i i i 1,p , x > 0, (bj , Bj , βj )1,q (α − 1, 1, 1) p,q where
m,n+1 k∗−1 = a−α Ip+1,q
"
δ aσ
# (1 − α, σ, 1) (ai , Ai , αi )1,p . (bj , Bj , βj )1,q
Therefore the distributions with pdf (4.9)-(4.11) are also the particular cases of the I-FIG distribution. References [1]
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X. Minghua, W. Yik-Chung, and A. Sonia. “Exact outage probability of dual-hop CSI-assisted AF relaying over Nakagami-m fading channels”. In: IEEE Transactions on Signal Processing 60.10 (2012), pp. 5578–5583. J. Prathima, T. M. V. Nambisan, and K. S. Kumari. “A study of I-function of several complex variables”. In: International Journal of Engineering Mathematics (2014). A. K. Rathie. “A new generalization of generalized hypergeometric functions”. In: Le Mathematiche 52 (1997), pp. 297–310. N. Sebastian. “A generalized gamma model associated with a Bessel function”. In: Integral Transforms and Special Functions 22.9 (2011), pp. 631–645. M. D. Springer. The Algebra of Random Variables. New York: John Wiley, 1979. M. D. Springer and W. E. Thompson. “The distribution of products of beta, gamma and Gaussian random variables”. In: SIAM J. Appl. Math. 18.4 (1970), pp. 721–737. E. C. Titchmarsh. Introduction to the Theory of Fourier Integrals. Oxford: Oxford University Press, 1937.
P. Vellaisamy, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, INDIA. E-mail address:
[email protected] Kuldeep Kumar Kataria, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, INDIA. E-mail address:
[email protected]
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