Analytical formulation of the single-visit completeness joint   probability density function
  We derive an exact formulation of the multivariate integral representing the single-visit obscurational and photometric completeness joint probability density function for arbitrary distributions for planetary parameters. We present a derivation of the region of nonzero values of this function which extends previous work, and discuss time and computational complexity costs and benefits of the method. We present a working implementation, and demonstrate excellent agreement between this approach and Monte Carlo simulation results 
Daniel Garrett and Dmitry Savransky
astro-ph.IM astro-ph.EP