The distributions of ratios of random variables arise in many applied problems such as in structural dynamics working with frequency response functions (FRFs) and transmissibility functions (TFs). When analysing the distribution properties of ratio random variables through the definition of probability density functions (PDF), the problem is usually accompanied by multiple integrals. In this study, a unified solution is presented to efficiently calculate the PDF of a ratio random variable with its denominator and numerator specified by arbitrary distributions. With the use of probability density transformation principle, a unified expression can be derived for the ratio random variable by reducing the concerned problem into two-dimensional integrals. As a result, the PDFs of the ratio random variable can be efficiently computed by using effective numerical integration techniques. Finally, based on the vibration tests performed on the Alamosa Canyon Bridge, the proposed method is applied to the data to quantify the uncertainty of FRFs and TFs.