We give a new model of learning motivated by smoothed analysis (Spielman and Teng, 2001). In this model, we analyze two new algorithms, for PAC-learning DNFs and agnostically learning decision trees, from random examples drawn from a constant-bounded product distributions. These two problems had previously been solved using membership queries (Jackson, 1995; Gopalan et al, 2005). Our analysis demonstrates that the "heavy" Fourier coefficients of a DNF suffice to recover the DNF. We also show that a structural property of the Fourier spectrum of any boolean function over "typical" product distributions. In a second model, we consider a simple new distribution over the boolean hypercube, one which is symmetric but is not the uniform distribution, from which we can learn O(log n)-depth decision trees in polynomial time.