Sums, products, and ratios for Freund’s bivariate exponential distribution

Abstract

We derive the exact distributions of R = X + Y, P = XY and W = X/(X + Y) and the corresponding moment properties when X and Y follow Freund’s bivariate exponential distribution. The expressions turn out to involve special functions. We also provide extensive tabulations of the percentage points associated with the distributions. These tables—obtained using intensive computing power—will be of use to practitioners of the bivariate exponential distribution.

Introduction

Since the 1930s, the statistics literature has seen many developments in the theory and applications of linear combinations and ratios of random variables. Some of these include:

• Ratios of normal random variables appear as sampling distributions in single equation models, in simultaneous equations models, as posterior distributions for parameters of regression models and as modeling distributions, especially in economics when demand models involve the indirect utility function (details in [25]).

• Weighted sums of uniform random variables—in addition to the well known application to the generation of random variables—have applications in stochastic processes which in many cases can be modeled by these weighted sums. In computer vision algorithms these weighted sums play a pivotal role [9]. An earlier application of the linear combinations of uniform random variables is given in connection with the distribution of errors in nth tabular differences Δⁿ [12].

• Ratio of linear combinations of chi-squared random variables are part of von Neumann [14] test statistics (mean square successive difference divided by the variance). These ratios appear in various two-stage tests [23]. They are also used in tests on structural coefficients of a multivariate linear functional relationship model (details in [3], [20]).

• Sums of independent gamma random variables have applications in queuing theory problems such as determination of the total waiting time and in civil engineering problems such as determination of the total excess water flow into a dam. They also appear in test statistics used to determine the confidence limits for the coefficient of variation of fiber diameters [11], [8] and in connection with the inference about the mean of the two-parameter gamma distribution [7].

• Linear combinations of inverted gamma random variables are used for testing hypotheses and interval estimation based on generalized p-values, specifically for the Behrens–Fisher problem and variance components in balanced mixed linear models [24].

• As to the Beta distributions their linear combinations occur in calculations of the power of a number of tests in ANOVA [13] among other applications. More generally, the linear combinations are used for detecting changes in the location of the distribution of a sequence of observations in quality control problems [10]. Pham-Gia and Turkkan [16], [17], [18], [19] and Pham-Gia [15] provided applications of sums and ratios to availability, Bayesian quality control and reliability.

• Linear combinations of the form $T=a_1{t}_{f_1}+a_2{t}_{f_2}$, where t_f denotes the Student t random variable based on f degrees of freedom, represents the Behrens–Fisher statistic and—as early as the middle of the twentieth century—Stein [22] and Chapman [2] developed a two-stage sampling procedure involving the T to test whether the ratio of two normal random variables is equal to a specified constant.

• Weighted sums of the Poisson parameters are used in medical applications for directly standardized mortality rates [4].

In this paper, we consider the distributions of R = X + Y, P = XY and W = X/(X + Y) when X and Y are correlated exponential random variables with the joint pdf given by

$$f(x\text{,}y)=\left\{\begin{array}{cc}\alpha {\beta }^{\prime }\exp \{-{\beta }^{\prime }y-(\alpha +\beta -{\beta }^{\prime })x\}\text{,}\hfill & \mathrm{if}0<x<y\text{,}\hfill \\ {\alpha }^{\prime }\beta \exp \{-{\alpha }^{\prime }x-(\alpha +\beta -{\alpha }^{\prime })y\}\text{,}\hfill & \mathrm{if}0<y⩽x\hfill \end{array}\right.$$

for x > 0, y > 0, α > 0, β > 0, α′ > 0 and β′ > 0. This distribution is due to Freund [5] and therefore known as Freund’s bivariate exponential distribution. It has received applications in several areas especially in reliability. For example, Barlow and Proschan [1] applied this distribution to data on failures of caterpillar tractors.

The paper is organized as follows. In Sections 2 PDFS, 3 Moments, we derive explicit expressions for the pdfs and moments of R = X + Y, P = XY and W = X/(X + Y). In Section 4, we provide extensive tabulations of the associated percentage points, obtained by means of intensive computing power. These values will be of use to the practitioners of the bivariate exponential distribution.

The calculations of this paper involve the complementary incomplete gamma function defined by

$$\Gamma (a\text{,}x)=\underset{x}{\overset{\infty }{\int }}{t}^{a-1}\exp (-t)\mathrm{d}t\text{.}$$

The properties of this special function can be found in [21], [6].