Numerous decision problems are solved using the tools of distributionally robust optimization. In this framework, the distribution of the problem's random parameter is assumed to be known only partially in the form of, for example, the values of its first moments. The aim is to minimize the expected value of a function of the decision variables, assuming the worst-possible realization of the unknown probability measure. In the general moment problem approach, the worst-case distributions are atomic. We propose to model smooth uncertain density functions using sum-of-squares polynomials with known moments over a given domain. We show that in this setup, one can evaluate the worst-case expected values of the functions of the decision variables in a computationally tractable way.