Scientific Journal of Information Engineering October 2015, Volume 5, Issue 5, PP.155-158
Some Moments of Linear Combination of Marshall and Olinkin’s Biviariate Exponential Distribution Yunfei Guo# Mathematics Department, Yanbian University, Yanji 133002, China #
Email: guoyunfei0413@sina.com
Abstract In this paper, explicit derivation of moments of X and Y which follow Marshall and Olkin’s bivariate exponential distribution (MOBVE) with dependence between them first. Furthermore, we derive the moment properties of Z = α X + β Y (α > 0, β > 0 ) when ( X , Y ) follow the above MOBVE. Keywords: Moment; Linear Combination; Bivariate Exponential Distribution
1 INTRODUCTION Bivariate exponential distributions are one of the most important distributions in the area of reliability. When there are two or more variables affecting the system, in most of the cases the analysis is carried out by assuming that they are statistically independent. However, the assumption of independence does not hold sometime in practice. Several bivariate models have been introduced in the literature. Some well know bivariate exponential distributions are those by Gumbel[1](1960), Freund[2](1961), Marshall and Olkin[3](1967), Block and Basu[4](1974),Downton[5](1970) and so on. And these distributions attracted many practical applications in reliability problems. Besides, the distribution of Z = α X + β Y (α > 0, β > 0 ) which is of interest in quality and reliability engineering, has been studied by several authors especially when X and Y are independent random variables. Moments of X and Y which follow MOBVE has been studied in Cheng[5],Saralees[6]. Saralees also studied moment properties of R= X + Y which follow Freund’s bivariate exponential distribution[7], bivariate Gumbel distribution[8] and Lawrance and Lewis’s bivariate exponential distribution[9]. This paper proves some moments of X and Y which follow MOBVE. Explicit expressions of some moments of Z = α X + β Y (α > 0, β > 0 ) are also provided in this paper. First of all, some necessary pre-knowledge are given in the following section.
2 PRE-KNOWLEDGE The bivariate exponential distribution in this paper refers to Marshall and Olkin’s bivariate exponential distribution (MOBVE ( λ1 , λ2 , λ12 )) as follows. Definition 2.1[6] Marshall and Olkin’s bivariate exponential distribution (MOBVE ( λ1 , λ2 , λ12 )) has the joint pdf specified by λ ( λ + λ12 ) exp {−λ1 x − ( λ2 + λ12 ) y} ,if x < y 1 2 f ( x, = y ) λ2 ( λ1 + λ12 ) exp {−λ2 y − ( λ1 + λ12 ) x} ,if x > y λ12 exp {− ( λ1 + λ2 + λ12 ) y} ,if x = y
(2.1)
for x > 0, y > 0, λ1 > 0, λ2 > 0, λ12 > 0 . This distribution arises in the following context: X and Y are the lifetimes of two components subjected to three kinds of shocks; these shocks are assumed to be - 155 http://www.sjie.org
governed by independent Poisson processes with parameters λ1 , λ2 and λ12 ,according as the shock applies to component 1 only, component 2 only, or both components. The distributions has received wide applicability in nuclear reactor safety; competing risks, reliability and in quantal response contexts. Besides, some moment formulas can be obtained easily from reference[5]. E(X ) = Var ( X ) =
1 1 , E (Y ) = λ1 + λ12 λ2 + λ12 1
( λ1 + λ12 )
, Var (Y ) =
1
(2.3)
( λ2 + λ12 )2
1 1 1 + λ λ1 + λ12 λ2 + λ12
E ( XY ) =
The correlation coefficient = ρ Corr = ( X ,Y )
2
(2.2)
(2.4)
λ12 , where λ λ = λ1 + λ2 + λ12
(2.5)
λ12 λ is written to ρ Cov = = ( X , Y ) 12 . λ λ
Note: in the reference[5], the ρ Corr = = ( X ,Y )
However, Cov ( X , Y ) denotes to the covariance rather than the correlation coefficient, which will be calculated in the paper. Lemma 2.1[10] If X and Y are jointly distributed according to (2.1), then we can obtain that
(
j −1 Γ( j + k) Γ (i + k ) + iΓ ( j + 1) ∑ i k j k − + j −k λ j +k Γ ( k + 1) λ1 λ 0= k 0 Γ ( k + 1) λ2
i −1
)
E X iY j = jΓ ( i + 1) ∑
k =
(2.6)
λ + λ , i and j are positive integers. where λ = λ1 + λ2 + λ12 , λ= 1 λ1 + λ12 , λ= 2 2 12 To obtain expectation, variance and other higher moments of Z = α X + β Y (α > 0, β > 0 ) , some moment formulas such as (2.2), (2.3), (2.4) and (2.5) have been given above. However, in these formulas, explicit derivation of E ( XY ) , Cov ( X , Y ) and ρ = Corr ( X , Y ) have not provided in reference [5], so we will give the detailed process of deriving from lemma 2 to lemma 4, using a different method from that in reference[10]. Lemma 2.2 If X and Y are jointly distributed according to (2.1), then E ( XY ) =
1 1 1 + . λ λ1 + λ12 λ2 + λ12
Proof. First method can be found in reference[11]. Now we give another method to prove it.If i= j= 1 in Lemma 1, we can see that, 0
0 Γ (1 + k ) Γ (1 + k ) + Γ (1 + 1) ∑ 1 1 − k + k 1− k λ1+ k Γ ( k + 1) λ λ 0= k 0 Γ ( k + 1) λ
E ( XY ) = Γ (1 + 1) ∑
= k
= Γ ( 2)
1
2
Γ (1) Γ (1) 1 1 1 1 1 1 + Γ ( 2) = + = + Γ (1) λ1λ Γ (1) λ2 λ λ λ1 λ1 λ λ1 + λ12 λ2 + λ12
Lemma 2.3 If X and Y are jointly distributed according to (2.1), then Cov ( X , Y ) =
λ12 λ ( λ1 + λ12 )( λ2 + λ12 )
Cov ( X , Y= ) E ( XY ) − EX ⋅ EY
Proof. =
1
1
λ λ1 + λ12
+
1
−
1
⋅
1
=
λ2 + λ12 λ1 + λ12 λ2 + λ12
λ2 + λ12 + λ1 + λ12 − λ λ12 = λ ( λ1 + λ12 )( λ2 + λ12 ) λ ( λ1 + λ12 )( λ2 + λ12 )
- 156 http://www.sjie.org
Lemma 2.4 If X and Y are jointly distributed according to (2.1), then ρ Corr = = ( X ,Y )
Proof.
λ12 λ
Cov ( X , Y ) λ12 ( λ1 + λ12 )( λ2 + λ12 ) λ12 = = ρ Corr = = ( X ,Y ) σ ( X ) σ (Y ) λ ( λ1 + λ12 )( λ2 + λ12 ) λ
3 MOMENT PROPERTIES OF
Z = α X + β Y (α > 0, β > 0 )
In this section, we will derive some moments of Z = α X + β Y (α > 0, β > 0 ) , including expectation, variance and higher moments in the following theorems. Theorem 3.1 If X and Y are jointly distributed according to (2.1), then we can obtain that
αλ2 + βλ1 + (α + β ) λ12 ( λ1 + λ12 )( λ2 + λ12 )
EZ= E (α X + β Y )=
Proof.
EZ = E (α X + β Y ) = α EX + β EY =
Collary 3.1 If α =β =1, EZ =E ( X + Y ) =
α
+
λ1 + λ12
β λ2 + λ12
=
(3.1)
αλ2 + βλ1 + (α + β ) λ12 ( λ1 + λ12 )( λ2 + λ12 )
λ + λ12 ( λ1 + λ12 )( λ2 + λ12 )
Theorem 3.2 If X and Y are jointly distributed according to (2.1), then we can obtain that
α 2 λ ( λ2 + λ12 ) + β 2 λ ( λ1 + λ12 ) + 2αβλ12 ( λ1 + λ12 )( λ2 + λ12 ) 2
DZ == Proof.
2
λ ( λ1 + λ12 ) ( λ2 + λ12 ) 2
(3.2)
2
DZ= D (α X + β Y )= α 2 DX + β 2 DY + 2αβ Cov(X, Y)
=α 2
1
( λ1 + λ12 )
2
+ β2
1
( λ2 + λ12 )
2
λ12 λ ( λ1 + λ12 )( λ2 + λ12 )
α 2 λ ( λ2 + λ12 ) + β 2 λ ( λ1 + λ12 ) + 2αβλ12 ( λ1 + λ12 )( λ2 + λ12 ) 2
=
+ 2αβ
2
λ ( λ1 + λ12 ) ( λ2 + λ12 ) 2
2
Collary 3.2 If α= β= 1, we can easily obtained that
DZ= D ( X + Y = )
λ ( λ2 + λ12 ) + λ ( λ1 + λ12 ) + 2λ12 ( λ1 + λ12 )( λ2 + λ12 ) 2
2
λ ( λ1 + λ12 ) ( λ2 + λ12 ) 2
2
Theorem 3.3 If X and Y are jointly distributed according to (2.1), then we can obtain that
( )
E Zn
n −l −1 Γ (l + k ) l Γ ( n − l + 1) ∑ n −l − k l + k n λ k = 0 Γ ( k + 1) λ1 == ∑ n α n −l β l l −1 l Γ − + n l k) ( l =0 + ( n − l ) Γ ( l + 1) ∑ l −k λ l +k k = 0 Γ ( k + 1) λ2
()
( )
()
(
)
()
(3.3)
(
)
n n Proof. The result in (3.3) follows by writing E Z n = ∑ n E α n −l X n −l β l Y l = ∑ n α n −l β l E X n −l Y l , l l l =0 l =0
and applying Lemma 2.1 to each expectations in the sum.
4 CONCLUSION Explicit derivation of moments of X and Y which follow Marshall and Olkin’s bivariate exponential distribution - 157 http://www.sjie.org
(MOBVE) with dependence between them first.Besides,we provide the moment properties of= Z α X + β Y when X , Y follow the above MOBVE. Moment properties of other bivariate exponential distributions will be studied ( ) further in future.
REFERENCES [1]
Gumbel, E. J., 1960. Bivariate exponential distributions. J. Amer. Statist. Assoc. 55 (292), 698-707.
[2]
Freund. J.E., 1961. A bivariate extension of the exponential distribution. J.Amer. Statist.Assoc. 56(296), 971-977.
[3]
Marshall,A.W., Olkin,I., 1967. A multivariate exponential distribution. J.Amer. Statist. Assoc. 62(317), 30-44.
[4]
Block, H.W., Basu, A.P., 1974. A continuous, bivariate exponential extension. J.Amer. Statist.Assoc. 69(348), 1031-1037.
[5]
Cheng Kan, Cao Jinhua, 2006. Introduction to reliability mathematics (the revised edition) [M]. Beijing,Higher Education Press,.
[6]
Saralees Narajah and Samuel Kotz, 2006. Reliability for some Bivariate exponential distributions, Mathematical Problems in Engineering, 1-14.
[7]
Arjun K. Gupta, Saralees Nadarajah, 2006. Sums, products, and ratios for Freund’s bivariate exponential distribution, Applied Mathematics and Computation 173, 1334–1349.
[8]
Saralees Nadarajah, 2005. Sums, Products, and Ratios for the Bivariate Gumbel Distribution, Mathematical and Computer Modeling 42,499-518.
[9]
Saralees Nadarajah, M. Masoom Ali, 2006. The distribution of sums, products and ratios for Lawrance and Lewis’s bivariate exponential random variables, Computational Statistics & Data Analysis 50, 3449 – 3463.
[10] Albert W. Marshall and Ingram Olkin, 1967. A Multivariate Exponential Distribution, Journal of the American Statistical Association, Vol. 62, 30-44.
AUTHORS 1
Yunfei Guo was born on April 13th, 1983 in Jilin province, and received his M.S. degrees in Yanbian University,
China in 2010. He is a Lecture of Yanbian University. His research interests are reliability and statistical analysis.
- 158 http://www.sjie.org