# On the Linear Combination of Exponential and Gamma Random Variables

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## Abstract

**:**

## 1 Introduction

- In automatic control, one often encounters the problem of maximizing the expected sum of n variables, chosen from a sequence of N sequentially arriving i.i.d. scalar random variables, X
_{1}, X_{2}, . . . , X_{N}. The objective is to devise a decision rule so as to maximize $\sum}_{i=1}^{n}{X}_{{k}_{i}$, where k_{i}∈ {1, 2, . . . , N} is the index of the ith random variable selected. At time k, the random variable X_{k}is observed, and the decision to select the value or not must be taken online. This problem is known as the sequential screening problem and many decision problems can be formulated in this way (Pronzato [1]). - The theory of congruence equations (see, for example, Cerruti [2]) has applications in computer science. There is a wide literature about congruence equations and the last twenty years has seen interesting formulas and functions derived: among these, expressions giving the number of solutions of linear congruences. Counting such solutions has relations with statistical problems like the distribution of the values taken by particular sums.
- In neurocomputing, linear combinations are used for combining multiple probabilistic classifiers on different feature sets. In order to achieve the improved classification performance, a generalized finite mixture model is proposed as a linear combination scheme and implemented based on radial basis function networks. In the linear combination scheme, soft competition on different feature sets is adopted as an automatic feature rank mechanism so that different feature sets can be always simultaneously used in an optimal way to determine linear combination weights (Chen and Chi [3]).

## 2 PDF and CDF

**Theorem 1**

**Proof:**

**Corollary 1**

**Corollary 2**

**Corollary 3**

**Corollary 4**

## 3 Entropy

cc<-lambda*((mu*alpha)**a)/(alpha*gamma(a)*(mu*alpha-lambda*beta)**a) ff<-function (x) {tt<-gamma(a)*pgamma(x*(mu*alpha-lambda*beta)/(alpha*beta),shape=a) tt<-exp(-lambda*x/alpha)*tt*log(tt) return(tt)} ent<-1+lambda*beta*a/(alpha*mu)-log(cc) ent<-ent-cc*integrate(ff,lower=0,upper=Inf)$value

## 4 Percentiles

_{p}associated with the cdf of Z = αX + βY. These percentiles are computed numerically by solving the equations

_{p}for p = 0.01, 0.05, 0.1, 0.90, 0.95, 0.99 for given values of α, β, λ, µ and a.

#this program gives percentiles when beta > 0 ff:=(1/GAMMA(a))*((mu*alpha)/(mu*alpha-lambda*beta))**a*exp(-lambda*z/alpha): ff:=ff*(GAMMA(a)-GAMMA(a,z*(mu*alpha-lambda*beta)/(alpha*beta))): ff:=1-GAMMA(a,mu*z/beta)/GAMMA(a)-ff: p1:=fsolve(ff=0.01,z=0..1000): p2:=fsolve(ff=0.05,z=0..1000): p3:=fsolve(ff=0.1,z=0..1000): p4:=fsolve(ff=0.90,z=0..1000): p5:=fsolve(ff=0.95,z=0..1000): p6:=fsolve(ff=0.99,z=0..1000): print(p1,p2,p3,p4,p5,p6); #this program gives percentiles when beta < 0 ff1:=(1/GAMMA(a))*((mu*alpha)/(mu*alpha-lambda*beta))**a: ff1:=ff1*exp(-lambda*z/alpha): ff1:=ff1*GAMMA(a,z*(mu*alpha-lambda*beta)/(alpha*beta)): ff1:=GAMMA(a,mu*z/beta)/GAMMA(a)-ff1: ff2:=1-((mu*alpha)/(mu*alpha-lambda*beta))**a*exp(-lambda*z/alpha): bd:=1-((mu*alpha)/(mu*alpha-lambda*beta))**a: if (bd>0.01) then p1:=fsolve(ff1=0.01,z=-1000..0): end if: if (bd<=0.01) then p1:=fsolve(ff2=0.01,z=0..1000): end if: if (bd>0.05) then p2:=fsolve(ff1=0.05,z=-1000..0): end if: if (bd<=0.05) then p2:=fsolve(ff2=0.05,z=0..1000): end if: if (bd>0.1) then p3:=fsolve(ff1=0.1,z=-1000..0): end if: if (bd<=0.1) then p3:=fsolve(ff2=0.1,z=0..1000): end if: if (bd>0.9) then p4:=fsolve(ff1=0.9,z=-1000..0): end if: if (bd<=0.9) then p4:=fsolve(ff2=0.9,z=0..1000): end if: if (bd>0.95) then p5:=fsolve(ff1=0.95,z=-1000..0): end if: if (bd<=0.95) then p5:=fsolve(ff2=0.95,z=0..1000): end if: if (bd>0.99) then p6:=fsolve(ff1=0.99,z=-1000..0): end if: if (bd<=0.99) then p6:=fsolve(ff2=0.99,z=0..1000): end if: print(p1,p2,p3,p4,p5,p6);We hope these programs will be of use to the practitioners of the linear combination (see Section 1).

## Acknowledgments

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**Figure 1.**Plots of the pdf of (3)–(4) for λ = 1, µ = 1, a = 0.5, 2, 5, 10, and (a): α = 1 and β = 1; (b): α = 1 and β = −1; (c): α = 1 and β = 2; and, (d): α = 1 and β = −2. The curves with the left to the right correspond to increasing values of a.

**Figure 2.**Plots of the Shannon entropy for λ = 1, µ = 1, β = 1, α = 2, 3, . . . , 10 and a = 0.1, 0.2, . . . , 10 (left); for λ = 1, µ = 1, α = 2, β = −1, −2, . . . , −9 and a = 0.1, 0.2, . . . , 10 (right). The curves in the top plot from the bottom to the top correspond to increasing values of α. The curves in the bottom plot from the left to the right correspond to increasing values of β.

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**MDPI and ACS Style**

Nadarajah, S.; Kotz, S.
On the Linear Combination of Exponential and Gamma Random Variables. *Entropy* **2005**, *7*, 161-171.
https://doi.org/10.3390/e7020161

**AMA Style**

Nadarajah S, Kotz S.
On the Linear Combination of Exponential and Gamma Random Variables. *Entropy*. 2005; 7(2):161-171.
https://doi.org/10.3390/e7020161

**Chicago/Turabian Style**

Nadarajah, Saralees, and Samuel Kotz.
2005. "On the Linear Combination of Exponential and Gamma Random Variables" *Entropy* 7, no. 2: 161-171.
https://doi.org/10.3390/e7020161