A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one
  We prove that, when all elliptic curves over $\mathbb{Q}$ are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are both strictly positive. 
Manjul Bhargava and Christopher Skinner
math.NT