Mathematical Computation June 2015, Volume 4, Issue 2, PP.46-49
Study on the Linear Combination of a Class of Bivariate Exponential Distribution Yunfei Guo Mathematics Department, Yanbian University, Yanji 133002, China Email: guoyunfei0413@sina.com
Abstract Making use of an easier method, this paper derives the probability density function (pdf) of the linear combination of random variable= Z α X + β Y . Furthermore, cumulative distribution function (cdf) of= Z α X + β Y , where X and Y have the bivariate exponential distributions, has also been proposed in this paper. Results from this paper would have important role in the reliability study. Keywords: Linear Combination; Bivariate Exponential; Reliability
1 INTRODUCTION The distributions of the linear combination of two independent random variables arise in many fields of research, see, for example, Ladekarl[1] ,Amari and Misra[2], Cigizoglu and Bayazit[3], Galambos and Simonelli[4], Nadarajah and Kibria[5], among others. In recent years, there has been a great interest in the study of the distributions of the linear combination α X + β Y ,when X and Y are independent random variables and belong to different families, among them, Nadarajah and Kotz[6] for the linear combination of exponential and gamma random variables, Kibria and Nadarajah[7] for the linear combination of exponential and Rayleigh random variables, and Nason[8] for the distributions of the sum X + Y , when X and Y are independent normal and sphered student’s t random variables respectively, are notable. Recently, Guo[9] provides the exact distributions of the linear combination of the bivariate exponential distributions, and Zhang[10] gives some revise to the results of Guo’s. The method in Zhang’s paper of deriving the the probability density function (pdf) of the random variable= Z α X + β Y , however, is complex in some way. Based on this, this paper provides a pretty simpler way to derive the probability density function (pdf) of the random variable= Z α X + β Y . Besides, further research has been finished with the conclusion above. The organization of this paper is as follows. In section 2, the necessary pre-knowledge is prepared for the following work. In section 3, the pdf of= Z α X + β Y has been derived in a more simple way, and cumulative distribution function (cdf) of= Z α X + β Y has been also provided. Section 4 concludes the paper and discusses possible future work.
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. Definition2.1[11] Marshall and Olkin’s bivariate exponential distribution (MOBVE ( λ1 , λ2 , λ12 )) has the joint pdf specified by λ1 ( λ2 + λ12 ) exp {−λ1 x − ( λ2 + λ12 ) y} ,if x < y f ( x, = y ) λ2 ( λ1 + λ12 ) exp {−λ2 y − ( λ1 + λ12 ) x} ,if x > y λ12 exp {− ( λ1 + λ2 + λ12 ) y} ,if x = y for x > 0, y > 0, λ1 > 0, λ2 > 0, λ12 > 0 . This distribution arises in the following context:
(2.1)
X and Y are the lifetimes of two components subjected to three kinds of shocks; these shocks are assumed to be - 46 www.ivypub.org/mc
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. Proposition2.2[10] Let f ( x, y ) denotes the joint probability density function of bivariate continuous random variable Z α X + β Y is ( X , Y ) , then pdf of= fZ ( z ) =
1
β
Where α ≠ 0, β ≠ 0 , −∞ < z < +∞
1 z −αx dx = α
+∞ ∫−∞ f x, β
z−βy dy
+∞ ∫−∞ f x, α
(2.2)
The proof of this result can be referred in the reference[12]. Theorem 2.3 If X and Y have the joint probability density function (2.1), pdf of= Z α X + β Y is: λ1 ( λ2 + λ12 ) λ1 + λ2 + λ12 λ + λ12 z − exp − 2 z + exp − α +β β α ( λ2 + λ12 ) − βλ1 λ (λ + λ ) + + + λ λ λ λ λ 2 1 12 fZ ( z ) = 2 12 exp − 1 12 z − exp − 1 z , z > 0 αλ2 − β ( λ1 + λ12 ) α α +β 0, z ≤ 0
3 A NEW METHOD OF DERIVATION OF THE PDF OF= Z α X + βY Theorem3.1 Let f ( x, y ) denotes the joint probability density function of bivariate Z α X + β Y is ( X , Y ) , then pdf of= fZ ( z ) =
Where α ≠ 0, β ≠ 0 , −∞ < z < +∞
1
1 z −αx dx = α
+∞ ∫ f x, β β −∞
continuous random variable
z−βy dy
+∞ ∫−∞ f x, α
(2.3)
(3.1)
Note: proof of the theorem 3.1 in this paper is different from that of the theorem 2.2, although results of them are the same. Proof. Let
= U1 U= X 1 ( X ,Y ) U= U X , Y = X +Y ( ) 2 2
(3.2)
or X= X= U1 Z = α X + βY = U2
(3.3)
X= h1 (U1 ,U 2= ) X= U1 Z − α X U 2 − αU1 , = Y h2 (U = = 1 ,U 2 )
(3.4)
inverse transform of which is:
β
and Jacobi determinant = J
∂x ∂x ∂u1 ∂u2 = ∂y ∂y ∂u1 ∂u2
∂
∂u1 ∂u1 u2 − α u1
β
∂u1
β
∂u1 ∂u2 1 0 α 1 1 , so joint pdf of U1 = X and u= = 2 − α u1 − ∂ β
β
∂u2
U 2 = Z is :
- 47 www.ivypub.org/mc
β
β
z −αx f x, β β 1
= h ( x, z ) f= h1 ( u1 , u 2 ) , h2 ( u1 , u 2 ) J
(3.5)
Therefore, pdf of= Z α X + β Y can be calculated as follows: +∞
f Z ( z ) ∫= h ( x, z ) dx = −∞
1
z −αx dx
+∞ ∫ f x, β β −∞
By symmetry, f Z ( z ) can be also written in the following form:
(3.6)
z−βy (3.7) dy Through comparing the proof between theorem2.2 and Theorem3.1, we can see that the proof in theorem3.1 is pretty simpler than that in theorem2.2 because of preventing from the complex calculation of the integral. fZ ( z ) =
+∞ ∫ f x, α α −∞
1
The pdf of= Z α X + β Y if X and Y have the joint probability density function (2.1) has been obtained in theorem 2.3. The cumulative distribution function (cdf) of= Z α X + β Y , however, has been not derived yet. So in this paper, we will finish this work. Theorem3.2 If X and Y have the joint probability density function (2.1),and let, λ = λ1 + λ2 + λ12 ,and λ= λ2 + λ12 the cdf of= Z α X + β Y is: 2
FZ ( z= ) +
λ λ2 λ1 α + β λ β − exp(− z ) + exp − 2 z λ α +β αλ2 − βλ1 λ2 β
λ2 λ1 α α + β λ2 λ1 α + β β λ2 λ1 α λ α +β λ exp − z) + − exp(− 1 z) + − + − (3.8) λ α λ αλ2 − βλ1 λ1 λ2 αλ2 − βλ1 λ1 α + β αλ2 − βλ1 λ
Proof: FZ ( z ) = ∫−∞ f Z ( z )dz = Α + Β z
Where
= Α
λ2 λ2 λ2 λ1 α + β λ2 λ1 λ β λ z = − − + − − − exp( z ) exp z z dz exp( ) exp − z ∫ λ α +β α +β αλ2 − βλ1 λ2 αλ2 − βλ1 0 β β
=
λ λ2 λ1 α + β λ β − exp(− z ) + exp − 2 λ α +β αλ2 − βλ1 λ2 β
z 0
λ2 λ1 α + β β z + − λ2 αλ2 − βλ1 λ
And = Β
λ2 λ1 λ1 λ2 λ1 α λ1 λ α +β λ z − − − = − − exp( ) exp z z dz exp( z) + exp − z) ∫ 0 α α λ αλ2 − βλ1 αλ2 − βλ1 λ1 α + β α + β =
z 0
λ2 λ1 α α + β λ2 λ1 α λ α +β λ − exp(− 1 z) + exp − z) + − α λ λ αλ2 − βλ1 λ1 α + β αλ2 − βλ1 λ1
So
FZ ( z ) = Α + Β = +
λ λ2 λ1 α + β λ β − exp(− z ) + exp − 2 λ α +β αλ2 − βλ1 λ2 β
z
λ2 λ1 α + β β λ2 λ1 α α + β λ2 λ1 α λ α +β λ − + exp − z) + − exp(− 1 z) + − α λ λ λ2 αλ2 − βλ1 λ1 αλ2 − βλ1 λ1 α + β αλ2 − βλ1 λ - 48 www.ivypub.org/mc
4 CONCLUSION This paper derives the probability density function (pdf) of the linear combination of random variable= Z α X + βY . The detailed process of derivation is also provided. Besides, cumulative distribution function (cdf) of= Z α X + βY , where X and Y have the bivariate exponential distributions, has been proposed in this paper. Findings from this paper would have some important role in the practice. Other properties of bivariate exponential distributions will be studied further in future.
REFERENCES [1]
Ladekarl, M., Jensen, V., and Nielsen, B. Total number of cancer cell nuclei and mitoses in breast tumors estimated by the optical
[2]
Amari, S. V., and Misra, R. B. Closed-form expressions for distribution of sum of exponential random variables. IEEE
[3]
Cigizoglu, H. K., and Bayazit, M. A generalized seasonal model for flow duration curve. Hydrological Processes, 14, 1053-1067,
[4]
Galambos, J., and Simonelli, I. Products of Random Variables – Applications to Problems of Physics and to Arithmetical
[5]
Nadarajah, S., and Kibria, B. M. G. On the ratio of generalized Pareto random variables. Stochastic Environmental Research &
[6]
Nadarajah, S., and Kotz, S. On the linear combination of exponential and gamma random variables. Entropy, 7, 161-171, 2005
[7]
Kibria, B. M. G., and Nadarajah, S. Reliability modeling: Linear combination and ratio of exponential and Rayleigh. IEEE
disector. Analytical and Quantitative Cytology and Histology, 19, 329-337, 1997 Transactions on Reliability, 46, 519-522, 1997 2000 Functions. Boca Raton: CRC Press, 2005 Risk Assessment, 206-212, 2006
Transactions on Reliability, 56, 102-105, 2007 [8]
Nason, G. On the sum of t and Gaussian random variables [J]. Statistics and Probability Letters, 76, 1280-1286, 2006
[9]
Yunfei Guo. Exact distributions for the linear combination of bivariate exponential components of dependent variable[J]. Mathematics in Practice and Theory. 40(16): 179-183, 2010
[10] Lili Zhang. Revise of Exact distributions for the linear combination of bivariate exponential components of dependent variable[J]. Journal of Xi’an Shiyou University (Natural Sciene). 27(3): 102-105, 2012 [11] Saralees Narajah and Samuel Kotz. Reliability for some Bivariate exponential distributions, Mathematical Problems in Engineering, 1-14, 2006
AUTHORS 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.
- 49 www.ivypub.org/mc