Products, Sums and Quotients of Upper Truncated Pareto Random Variables with an Application in Hydrology

Some hydrological quantities such as the rainfall intensity, which is defined as the quotient of the rainfall depth to the rainfall duration, are based on functions of random variables. At this point, the probability distribution of that quantity arises. Then one may take this distribution into account for the exact statistical inference without referring to a simulation study. There are a lot of works on the exact distributions of functions of random variables in the literature. One case is for the Pareto distributed random variables. Pareto distribution and its upper truncated version have many applications in hydrological modelling. In this paper, the exact distributions of the product, sum and quotient of two independently distributed upper truncated Pareto random variables are obtained. Although the probability density functions of the product and quotient are obtained in elementary mathematical functions, that for the sum is obtained in terms of a special function. Some characteristics of these functions such as moments and percentiles can be easily obtained. The distributions of the quotient and the sum are applied on a rainfall data set from hydraulic efficiency research of green roofs. The parameters are estimated by the method of maximum likelihood. The theoretical results of this paper may also be useful to other practitioners of the upper truncated Pareto distribution.