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ON THE LINEAR COMBINATION OF LAPLACE RANDOM VARIABLES
Published online by Cambridge University Press: 31 August 2005
Abstract
The distribution of the linear combination αX + βY is derived when X and Y are independent Laplace random variables. Extensive tabulations of the associated percentage points are also given. The work is motivated by examples in automation, control, fuzzy sets, neurocomputing, and other areas of informational sciences.
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- Research Article
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- Probability in the Engineering and Informational Sciences , Volume 19 , Issue 4 , October 2005 , pp. 463 - 470
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- © 2005 Cambridge University Press
1. INTRODUCTION
For given random variables X and Y, the distribution of linear combinations of the form αX + βY is of interest in problems in automation, control, fuzzy sets, neurocomputing, and other areas of informational sciences. The distribution of αX + βY has been studied by several authors, especially when X and Y are independent random variables and come from the same family. For instance, see Fisher [8] and Chapman [3] for the Student's t family, Christopeit and Helmes [4] for the normal family, Davies [5] and Farebrother [7] for the chi-squared family, Ali [2] for the exponential family, Moschopoulos [12] and Provost [15] for the gamma family, Dobson, Kulasmaa, and Scherer [6] for the Poisson family, Pham-Gia and Turkkan [14] and Pham and Turkkan [13] for the beta family, Kamgar-Parsi, Kamgar-Parsi, and Brosh [11] and Albert [1] for the uniform family, Hitezenko [9] and Hu and Lin [10] for the Rayleigh family, and Witkovský [17] for the inverted gamma family.
In this article, we study the distribution of αX + βY when X and Y are independent Laplace random variables with probability density functions (p.d.f.s)
and
respectively, for −∞ < x < ∞, −∞ < y < ∞, λ > 0, μ > 0, −∞ < θ < ∞, and −∞ < φ < ∞. We assume without loss of generality that α > 0. Extensive tabulations of the associated percentage points are also provided.
2. CDF
Theorem 1 derives explicit expressions for the cumulative distribution function (c.d.f.) of αX + βY in terms of elementary functions.
Theorem 1: Suppose X and Y are distributed according to (1) and (2), respectively. The c.d.f. of Z = αX + βY can be expressed as one of the following:
1. If β < 0 and φ < (z − θα)/β, then
2. If β < 0 and φ > (z − θα)/β, then
3. If β > 0 and φ < (z − θα)/β, then
4. If β > 0 and φ > (z − θα)/β, then
For (3)–(6), p = λ(z/α − θ) − λβφ/α and r = μφ − μ(z − θα)/β.
Proof: The c.d.f. F(z) = Pr(αX + βY ≤ z) can be expressed as
where F(·) inside the integral denotes the c.d.f. corresponding to (1) and given by
The results in (3)–(6) follow by elementary integration of (7). █
The following corollaries provide the c.d.f.s for the sum and the difference of Laplace random variables.
Corollary 1: Suppose X and Y are distributed according to (1) and (2), respectively. Then the c.d.f. of Z = X + Y can be expressed as one of the following:
1. If φ < z − θ, then
2. If φ > z − θ, then
In the above, p = λ(z − θ − φ) and r = μ(θ + φ − z).
Proof: Set α = 1 and β = 1 into (5) and (6). █
Corollary 2: Suppose X and Y are distributed according to (1) and (2), respectively. Then the c.d.f. of Z = X − Y can be expressed as one of the following:
1. If φ < θ − z, then
2. If φ > θ − z, then
In the above, p = λ(z + φ − θ) and r = μ(z + φ − θ).
Proof: Set α = 1 and β = −1 into (3) and (4). █
3. PERCENTILES
In this section, we provide tabulations of percentage points zp associated with the c.d.f. of Z = αX + βY. These values are obtained by numerically solving the equation F(zp) = p, where F is given by one of the eight formulas in the previous section. The computation involved is elementary and was performed using the algebraic manipulation package MAPLE. Table 1 provides the numerical values of zp for β = −5, −4.9, …, −0.1, 0.1, …, 4.9, 5, α = 1, λ = 1, μ = 1, θ = 0, and φ = 0. We hope that these values will be of use to the practitioners mentioned in Section 1. Similar tabulations could be easily derived for other values of α, β, λ, μ, θ, and φ. Sample programs are shown in the Appendix.
Percentage Points of Z
Acknowledgments
The authors would like to thank the editor for carefully reading the manuscript and for his great help in improving the article.
APPENDIX
The following programs in MAPLE can be used to generate tables similar to that presented in Section 3.
#this program gives percentiles when beta > 0
tt:=(mu/2)*exp(-mu*abs(y-phi)):
tt1:=(1/2)*exp(lambda*((z-beta*y)/alpha-theta)):
tt2:=1-(1/2)*exp(-lambda*((z-beta*y)/alpha-theta)):
ff:=int(tt*tt1,y=(z-alpha*theta)/beta..infinity):
ff:=ff+int(tt*tt2,y=-infinity..(z-alpha*theta)/beta):
p1:=fsolve(ff=0.6,z=-10000..10000):
p2:=fsolve(ff=0.7,z=-10000..10000):
p3:=fsolve(ff=0.8,z=-10000..10000):
p4:=fsolve(ff=0.90,z=-10000..10000):
p5:=fsolve(ff=0.95,z=-10000..10000):
p6:=fsolve(ff=0.99,z=-10000..10000):
print(beta,p1,p2,p3,p4,p5,p6);
#this program gives percentiles when beta < 0
tt:=(mu/2)*exp(-mu*abs(y-phi)):
tt1:=(1/2)*exp(lambda*((z-beta*y)/alpha-theta)):
tt2:=1-(1/2)*exp(-lambda*((z-beta*y)/alpha-theta)):
ff:=int(tt*tt2,y=(z-alpha*theta)/beta..infinity):
ff:=ff+int(tt*tt1,y=-infinity..(z-alpha*theta)/beta):
p1:=fsolve(ff=0.6,z=-10000..10000):
p2:=fsolve(ff=0.7,z=-10000..10000):
p3:=fsolve(ff=0.8,z=-10000..10000):
p4:=fsolve(ff=0.90,z=-10000..10000):
p5:=fsolve(ff=0.95,z=-10000..10000):
p6:=fsolve(ff=0.99,z=-10000..10000):
print(beta,p1,p2,p3,p4,p5,p6);
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