It is well known that the Fourier transform is a powerful analytical tool in studying the distribution of sums of independent random variables. In this paper it is pointed out that the Mellin transform is a natural analytical tool to use in studying the distribution of products and quotients of independent random variables. Formulae are given for determining the probability density functions of the product and the quotient $\frac{\xi}{\eta}$, where $\xi$ and $\eta$ are independent positive random variables with p.d.f.'s $f(x)$ and $g(y)$, in terms of the Mellin transforms $F(s) = \int_0^\infty f(x) x^{s-1} dx$ and $G(s) = \int_0^\infty g(y)y^{s-1} dy$. An extension of the transform technique to random variables which are not everywhere positive is given. A number of examples including Student's $t$-distribution and Snedecor's $F$-distribution are worked out by the technique of this paper.