On the cumulative distribution function of the sum and harmonic mean of two α - μ random variables with applications

The sum and harmonic mean of random variables (RVs) appear frequently in performance evaluation studies of wireless communications systems. In particular, the sum appears in the performance evaluation of maximal ratio combining (MRC) and equal gain combining (EGC) diversity receivers whereas the harmonic mean is encountered in performance evaluation studies of cooperative diversity networks with non-regenerative relays. In this study, the authors propose highly accurate finite-series approximations for the cumulative distribution function (CDF) of the sum and harmonic mean of two α-μ distributed RVs. The proposed approximations do not require neither a look-up table nor solving complicated transcendental equations, as is typically the case in the literature. The results reveal that the proposed approximations are very accurate over the typical range of interest when compared with the exact CDFs computed using numerical integration methods.