On the Convolution of Gamma Distributions

SOOCHOW JOURNAL OF MATHEMATICS Volume 31, No. 2, pp. 205-211, April 2005

ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS

MOHAMED AKKOUCHI

Abstract. In this paper, we give a formula for the distribution of the sum of n independent random variables with gamma distributions. A formula for such a sum was provided by Mathai (see [5]) in 1982. But it was complicated. In the paper [1], Jasiulewicz and Kordecki derived a formula for the particular case of independent random variables having Erlang distributions by using Laplace transform. Our method is based on elementary computations and our result (see Theorem 2.1 below) is expressed by means of the generalized beta function.

1. Introduction

The distribution of the sum of n independent exponentially distributed random variables with different parameters βi (i = 1, 2, . . . , n) is given in [2], [3], [4] and [6]. In the paper [1], H. Jasiulewicz and W. Kordecki have given the distributions of this sum without assuming that all the parameters βi are different. They reduced the problem to the one of finding the distribution of the sum of independent random variables having Erlang distributions. All of the above problems are the special cases of a sum of independent random variables with gamma distributions. In the paper [5], A.M. Mathai has provided a formula for such a sum, but as it was noticed in [1] and [6], this formula is complicated even in the case where the random variable are exponentially distributed. We point out that in the papers [1] and [6] the authors derived directly their results instead of using the results obtained in [5]. We point out also that the methods used by the authors in [1] to get their results are based on the use of Laplace transform. Received October 2, 2003. AMS Subject Classification. 60E05, 60G50. Key words. independent random variables, convolution, gamma distributions, generalized beta function.

205 206 MOHAMED AKKOUCHI

The aim of this note is a contribution to the investigations of the distribution of the sum of n independent random variables having gamma distributions. Our main result is presented in Section two (see Theorem 2.1 below). Our method is based on some elementary computations and our result is expressed by means of a multiple integral involving the generalized beta function.

2. The Result

In all this note, X1, . . . , Xn will be n independent random variables having gamma distributions. More precisely, we suppose that every Xi has a probability density function (pdf) fXi given by

αi βi αi−1 fXi (t) := t exp( tβi) χ[0,∞](t), (1) Γ(αi) − where Γ is the usual gamma function, χ[0,∞](t) = 1 if t > 0 and χ[0,∞](t) = 0 elsewhere, and the numbers αi, βi are positive for all i = 1, 2, . . . , n. We say that Xi has a gamma distribution γ(αi, βi) with parameters αi and βi for all i = 1, 2, . . . , n. We would like to ﬁnd the distribution of the random variable:

Sn := X + X + + Xn. (2) 1 2 · · · We recall that for every complex number z with positive real part, Γ(z) is given by ∞ − Γ(z) = tz 1 exp( t) dt. (3) − Z0 We recall that the beta function B(z1, z2) is given for all complex numbers z1, z2 with positive real parts by

1 − − B(z , z ) = tz1 1(1 t)z2 1 dt. (4) 1 2 − Z0 The following useful identities:

π 2 2z1−1 2z2−1 B(z1, z2) = 2 cos (t) sin (t) dt, (5) Z0 and Γ(z1) Γ(z2) B(z1, z2) = = B(z2, z1) (6) Γ(z1 + z2) ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS 207 are well known. For all integer n greater or equal to 2, we recall that the generalized beta function Bn is deﬁned for all complex numbers z1, z2, . . . , zn, with positive real parts by

Γ(z1) Γ(z2) Γ(zn) Bn(z1, z2, . . . , zn) = · · · . (7) Γ(z + z + + zn) 1 2 · · · The aim of this note is to prove the following result.

Theorem 1. The probability distribution function fSn of the random variable

Sn deﬁned by (2) is given by the formula

α1+···+αn−1 fSn (t) = Ct 1 1 −tCβ ,...,β (u1,...,un−1) e 1 n Bα ,...,α (u , . . . , un− )du dun− , (8) × · · · 1 n 1 1 1 · · · 1 Z0 Z0 for all t > 0 and fS (t) = 0 for all t 0, where n ≤ βα1 βα2 βαn C = 1 2 · · · n , (9) Γ(α + α + + αn) 1 2 · · · and

n−1 n−1 n−1

Cβ1,...,βn (u1, . . . , un−1) := β1 uj + βi(1 ui) uj + βn (1 un−1), (10) j i − j i − Y=1 X=2 Y= and

n−1 ··· − 1 α1+ +αj 1 αj+1−1 Bα1,...,αn (u1, . . . , un−1) := uj (1 uj) , (11) Bn(α1, . . . , αn) − jY=1 for all u , . . . , un− [0, 1]. 1 1 ∈ Proof. In order to prove the result above, we are led to compute the integrals

α1 αn β1 βn In(g) := · · · Γ(α1) Γ(αn) ∞ ∞ · · · α1−1 αn−1 −(β1x1+···+βnxn) g(x + + xn)x x e dx dxn, (12) × · · · 1 · · · 1 · · · n 1 · · · Z0 Z0 for all g b(R) the space of all continuous and bounded functions on the real ∈ C 2 line. To compute (12), we set xi := yi where 0 yi < for all i = 1, 2, . . . , n. ≤ ∞ 208 MOHAMED AKKOUCHI

With these new variables we have

n α1 αn 2 β1 βn In(g) = · · · Γ(α1) Γ(αn) ∞ ∞ · · · 2 2 2 2 2α1−1 2αn−1 −(β1y1 +···+βnyn ) g(y + +yn )y y e dy dyn. (13) × · · · 1 · · · 1 · · · n 1 · · · Z0 Z0 To compute the integral (13), we shall use the spherical coordinates. Thus, we set

y = r sin φn− sin φ sin φ sin φ 1 1 · · · 3 2 1 y = r sin φn− sin φ sin φ cos φ 2 1 · · · 3 2 1 y = r sin φn− sin φ cos φ 3 1 · · · 3 2 ......

yn−1 = r sin φn−1 cos φn−2

yn = r cos φn−1,

π where 0 r < , and 0 φk for all k = 1, . . . , n 1. Conversely, for all ≤ ∞ ≤ ≤ 2 − 2 2 2 k = 1, 2, . . . , n 1, by setting r := y + + y and rn := r, we have − k 1 · · · k yk+1 cos(φk) = , rk+1 and rk sin(φk) = . rk+1 We recall that the Jacobian of this change of variables is given by

n−1 n−2 n−3 r sin φn− sin φn− sin φ . (14) 1 2 · · · 2 With these spherical coordinates, we have

n n−1 2αi−1 2(α1+···+αn)−n 2(α1+···+αj )−j 2αj+1−1 yi = r sin cos . (15) i j Y=1 Y=1 We observe also that

2 2 2 2 β y + + βnyn = Cβ ,...,β (sin (φ ), . . . , sin (φn− )), (16) 1 1 · · · 1 n 1 1 ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS 209

where the function Cβ1,...,βn is deﬁned in (10). Taking into account the previous identities, we have

n α1 α ∞ π π β β n 2 2 2 2 2 2 1 n 2 2(α1+···+αn)−1 −r Cβ ,...,β (sin φ1,...,sin φn−1) In(g)= · · · g(r )r e 1 n Γ(α ) Γ(αn) 0 0 · · · 0 1 · · · Z "Z Z n−1 2(α1+···+αk)−1 2αk+1−1 sin φk cos φk dφ1 dφn−1 dr. (17) × " # · · · # kY=1 2 2 We set t = r and uj = sin φj for all j = 1, . . . , n 1. Then the Jacobian of this − mapping is given by − 1 n 1 1 . (18) n 2 √t j √uj 1 uj Y=1 − Therefore p βα1 βαn ∞ 1 1 1 n α1+···+αn−1 −tCβ ,...,β (u1,...,un−1) In(g)= · · · g(t)t e 1 n Γ(α ) Γ(αn) 0 0 · · · 0 1 · · · Z Z Z n−1 − α1+···+αk−1 αk+1 1 uk (1 uk) du1 dun−1 dt. (19) × " − # · · · # kY=1 Using the notations (11), the identity (19) gives

α1 α ∞ β β n ··· − I (g)= 1 · · · n g(t)tα1+ +αn 1 n Γ(α + +α ) 1 n Z0 1 1 · · · −tCβ ,...,β (u1,...,un−1) e 1 n Bα ,...,α (u , . . . , un− ) du dun− dt. (20) × · · · 1 n 1 1 1 · · · 1 Z0 Z0

(20) shows that the random variable Sn has a probability density function fSn given by fS (t) = 0 if t 0 and n ≤ α1 αn β1 βn α1+···+αn−1 fSn (t)= · · · t Γ(α1 + + αn) 1 1 · · · −tCβ ,...,β (u1,...,un−1) e 1 n Bα ,...,α (u , . . . , un− )du dun− , (21) × · · · 1 n 1 1 1 · · · 1 Z0 Z0 for all t > 0. Thus our theorem is completely proved.

n Remark 2. For all (α , . . . , αn) R , we have 1 ∈ 1 1 Bα ,...,α (u , . . . , un− ) du dun− = 1. (22) · · · 1 n 1 1 1 · · · 1 Z0 Z0 210 MOHAMED AKKOUCHI

n Proof. For all (α , . . . , αn) R , we have the following equalities 1 ∈ n−1 1 1 − α1+···+αk−1 αk+1 1 uk (1 uk) du1 dun−1 0 · · · 0 − · · · Z Z kY=1 n−1 1 − α1+···+αk−1 αk+1 1 = uk (1 uk) duk 0 − kY=1 Z n−1 = B(α + + αk, αk ) 1 · · · +1 kY=1 n−1 Γ(α + + αk)Γ(αk) Γ(α ) Γ(αn) = 1 · · · = 1 · · · . Γ(α1 + + αk+1) Γ(α1 + + αn) kY=1 · · · · · · Taking into account (7) and (11), our remark is proved.

We remark also that if all the βi are equal to a number β > 0, then for all u , . . . un [0, 1], we have 1 ∈

Cβ,...,β(u1, . . . , un−1) = β.

With the remarks made above, we have the following corollary.

Corollary 3. Suppose that all the βi are equal to a number β > 0, then the random variable Sn has a probability density function fSn given by fSn (t) = 0 if t 0 and ≤ α1+···+αn β α1+···+αn−1 −βt fSn (t) = t e , (23) Γ(α + + αn) 1 · · · for all t > 0.

Remark. (23) shows that under the assumptions of the previous corollary

Sn, has a gamma distribution γ(α + +αn, β) with parameters α + +αn and 1 · · · 1 · · · β. Thus we recapture in the previous corollary a well known result in Probabilities.

References

[1] H. Jasiulewicz and W. Kordecki, Convolutions of Erlang and of Pascal distributions with applications to reliability, Demonstratio Math., 36:1(2003), 231-238. [2] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions -1, Wiley, New York, 1994. [3] W. Kordecki, Reliability bounds for multistage structure with independent components, Stat. Probab. Lett., 34(1997), 43-51. ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS 211

[4] M. V. Lomonosov, Bernoulli scheme with closure, Problems Inform. Transmisssion, 10 (1974), 73-81. [5] A. M. Mathai, Storage capacity of a dam with gamma type inputs, Ann. Inst. Statist. Math., 34(1982), 591-597. [6] A. Sen and N. Balakrishnan, Convolutions of geometrics and a reliability problem, Stat. Probab. Lett., 43(1999), 421-426.

Département de Mathématiques, Université Cadi Ayyad, Faculté des Sciences-Semlalia, Bd. du Prince My. Abdellah BP 2390, Marrakech, Maroc (Morocco). E-mail: [email protected]