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A novel analytical framework for evaluating the statistics of the product of independent random variables is proposed. Compared with other methods, which use either infinite series or special functions, the new method provides simple and efficient closed-form approximations, in terms of elementary functions, such as powers and exponentials. Numerical results, which are used to check the accuracy of the new approximations, show that it is quite accurate in most regions of interest. As an application, the proposed analytical results can be efficiently used in wireless communications theory to evaluate in closed form the outage probability of cascaded fading channels, as well as the rate offset of the hybrid automatic repeat request (H-ARQ) transmission. Numerical examples show that the derived expressions provide significant insights on the behaviors of important system parameters as the outage probability and the rate offset.