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Stoch Environ Res Risk Assess (2007) 21:283286 DOI 10.1007/s00477-006-0063-4
ORIGINAL PAPER
Linear combination of Gumbel random variables
Saralees Nadarajah
Published online: 21 July 2006 Springer-Verlag 2006
Abstract The recent paper by Loaiciga and Leipnik (Stoch Environ Res Risk Environ 13:251259, 1999) derived the probability distribution of the sum of two independent Gumbel random variables. The results given are of little practical use because they are given in terms of characteristic functions. In this note, we consider the more general problem of deriving the linear combination of two independent Gumbel random variables. Explicit expressions are given for the probability density function and the cumulative distribution function of the linear combination. Various particular cases are also considered.
1 Introduction
The recent paper by Professors Loaiciga and Leipnik (1999) derived the probability distribution of the sum of two independent Gumbel random variables. The paper described several examples in hydrology where the distribution of the sum can be applied. However, the distribution is expressed in terms of characteristic functions and so its practical use is limited [because one would have to apply the inversion theorem to obtain the probability density function (pdf) and the cumulative distribution function (cdf)].
In this note, we consider the more general problem of the linear combination of two independent Gumbel random variables. We consider this general problem
because in hydrological sciences one encounters not just sums but also differences and other forms of linear combination. Some important examples are:
1. Stream ow from two rivers, or river basins, may combine to provide water supply to a single region. In this case, the total ow is the sum.
2. In a reservoirs catchment, the net in ow will be the difference between the areal precipitation and the amount of water taken out for purposes such as domestic use and agriculture.
3. Inter-arrival time of drought events is the sum of the drought duration and the successive non-drought duration.
Examples 1 and 3 involve the sum of two random variables while example 2 involves the difference. See Loaiciga and Leipnik (1999) for more examples on the sum of two random variables.
Let Z = a X + b Y denote the linear combination where X and Y are independent Gumbel random variables given by the pdfs