Approximations for the product, ratio, and sum ofα‐μrandom variables with application in the analysis of cognitive radio networks

Abstract

Summary Novel methods to approximate the probability density function of the product, ratio, and sum of α‐μ random variables are presented in this article. The approximations are simple to compute and produce results that range from fairly accurate to perfectly accurate, depending on the parameters of the involved α‐μ distributions. In addition, as an application of the results, it is investigated the performance of a cooperative cognitive network in which the secondary source and the relays are energy‐constrained nodes and harvest their energy from the primary network.

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

“…We mention here the interesting research related to non-negative random variables [35], continuous random variables [36], symmetric random variables [37] , subexponential random variables [38] or regular-variation-tailed random variables [39]. See also [40].…”

“…When one considers product random variables, the main attention is paid on their moments [28,35,57], probability distribution (i.e., PDF) [51,54,55] or other statistical characteristics [24,52]. However, there are also analyses related to tails behavior [2,38,39,47,58] or other asymptotic properties [40,53,59]. In this paper, we also study the basic characteristics of product random variables, like the PDF and moments, but we also examine their tail behavior.…”

“…Beyond theoretical considerations, the random variables that arise as a product (or other functions) of two variables have found various interesting applications including finance, economy, but also physical sciences, reliability theory, hydrology, and many others; see, e.g., [1,45,[62][63][64][65][66][67] and the corresponding comments in [2,[68][69][70]. The theory of product random variables was applied also in cognitive radio networks [40], sociological models [70], risk theory [2] or wireless communication systems [24]. In this paper, the proposed methodology is applied to the transaction values from the energy market.…”