We consider a model applicable in many communication systems where the sum of n stochastic sinusoidal signals of the same frequency, but with random amplitudes as well as phase angles is present. The exact probability distribution of the resulting signal's amplitude is of particular interest in many important applications, however the general problem has remained open. We derive new general formulae for the resultant amplitude's distribution, in terms of any given general joint density of the random variables involved. New exact probability densities of cases of interest follow from the general formulae which are applied to the problem of the exact evaluation of the bit error rate performance in narrowband multipath fading channels that consist of a small number of jointly dependent resolvable multipath components. The numerical results are compared with those of the currently popular method of Gaussian approximation. These are consistent with simulations, and show significant increase in accuracy particularly for small number of components.