BLOCK AND BASU'S BIVARIATE EXPONENTIAL DISTRIBUTION WITH APPLICATION TO DROUGHT DATA

1. INTRODUCTION

The aim of this article is to derive the exact distributions of R = X + Y , P = XY , and W = X /(X + Y ) when (X ,Y ) follows Block and Basu's [1] bivariate exponential distribution given by the joint probability density function (p.d.f.)

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for x > 0, y > 0, Α > 0, β > 0, Α' > 0, and β' > 0, where λ = λ₁ + λ₂ + λ₁₂ . This is one of the most flexible bivariate exponential distributions in the literature: It was derived by Block and Basu by omitting the singular part of Marshall and Olkin's [6] distribution.

Since the pioneering work of Gumbel [4], bivariate exponential distributions have attracted many applications in hydrological sciences. The above model due to Block and Basu [1] would be an ideal model for these applications. There is clear reason to believe that distributions of R = X + Y , P = XY , and W = X /(X + Y ) will be of interest in hydrological applications. For example, if X and Y denote the drought intensity and the drought duration, respectively, then P = XY will represent the magnitude of the drought. If X and Y denote the drought duration and the successive nondrought duration, respectively, then R = X + Y and W = X /(X + Y ) will represent the interarrival time of drought events and the proportion of drought events, respectively (see Section 4).

This article is organized as follows. In Sections 2 and 3 explicit expressions for the p.d.f.s and moments of R = X + Y , P = XY , and W = X /(X + Y ) are derived. In Section 4 an application of the results to drought data from Nebraska is provided. The calculations in this article involve the complementary incomplete gamma function defined by

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The properties of this special function can be found in Prudnikov, Brychkov, and Marichev [7] and Gradshteyn and Ryzhik [3].

2. PROBABILITY DENSITY FUNCTIONS

Theorems 1-3 derive the p.d.f.s of R = X + Y , P