we study some problems relating to polynomials evaluated either at random gaussian or random bernoulli inputs. we present a structure theorem for degree-d polynomials with gaussian inputs. in particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials $q_1,...,q_m$ so that the joint probability density function of $q_1(g),...,q_m(g)$ is close to being bounded. this says essentially that any abnormalities in the distribution of $p(g)$ can be explained by the way in which p decomposes into the $q_i$. we then present some applications of this result.