Non-central F distributions with non-centrality parameters with gamma distributions

The quotient of two linear combinations of independent chi-squares has a generalized F distribution. Exact expressions for these distributions, when the chi-squares are central and those in the numerator or in the denom- inator have even degrees of freedom, were given in Fonseca et al (2002). These expressions were extended for non-central chi-squares with random non-centrality parameters in Nunes and Mexia (2006). In this paper there will be considered random non-centrality parameters with Gamma distribu- tions, for the usual F distribution and for the generalized F distribution.