Statistics for Ratios of Rayleigh, Rician, Nakagami-m, and Weibull Distributed Random Variables

Abstract

The distributions of ratios of random variables are of interest in many areas of the sciences. In this brief paper, we present the joint probability density function (PDF) and PDF of maximum of ratiosμ1=R1/r1andμ2=R2/r2for the cases whereR1,R2,r1, andr2are Rayleigh, Rician, Nakagami-m, and Weibull distributed random variables. Random variablesR1andR2, as well as random variablesr1andr2, are correlated. Ascertaining on the suitability of the Weibull distribution to describe fading in both indoor and outdoor environments, special attention is dedicated to the case of Weibull random variables. For this case, analytical expressions for the joint PDF, PDF of maximum, PDF of minimum, and product moments of arbitrary number of ratiosμi=Ri/ri,i=1,…,Lare obtained. Random variables in numerator,Ri, as well as random variables in denominator,ri, are exponentially correlated. To the best of the authors' knowledge, analytical expressions for the PDF of minimum and product moments of{μi}i=1Lare novel in the open technical literature. The proposed mathematical analysis is complemented by various numerical results. An application of presented theoretical results is illustrated with respect to performance assessment of wireless systems.