The generalizedFvariate is the ratio of two independent gamma variates, and its distribution includes as special cases such distributions as the inverted beta, Lomax, and Snedecor'sF, Based on convolution, the distribution function of the sum of two independent generalizedFvariates is derived in terms of a Lauricella-Saran hyper-geometric function of three variables. The results are applied, with numerical examples given, to (a) a Bayesian analysis of the availability of a two-component series system and (b) a test of hypothesis on the multinormal mean vector whenever the covariance matrix has the intraclass correlation pattern.