My group is also leading studies to explore the architectures of exoplanetary systems: the mass- and orbital semi-major axis distributions and occurrence rates of exoplanets. We use different methods to characterize the frequency of different planets: for large-separation giant planets utilize direct imaging and for close-in planets we use planetary transits. We compare occurrence rates between different studies using statistical methods, including Monte Carlo simulations.
The low occurrence rate of giant planets around Sun-like stars
In Kasper, Apai et al. (2007) we carried out the first L’-band high-contrast imaging adaptive optics survey in search of giant exoplanets around the young stars in the Tuc-Hor and Beta Pictoris co-moving groups. We used the upper limits and compared these to the results of simulated observations of hypothetical planet populations, which were consistent with extrapolations of the short-period exoplanet populations as established by radial velocity surveys. The L’ imaging method and the Monte Carlo approach are now widely used.
More Planets around Lower-Mass Stars
In Mulders, Pascucci, Apai 2015a we modeled selection and detection biases for the Kepler mission to derive intrinsic planet-radius and semi-major axis distributions. This detailed statistical work led to the discovery of a stellar-mass dependent drop in 
exoplanet occurrence rates. Interestingly, the planet populations around stars with different masses show the same overall distribution, but – counterintuitively – less massive stars have more planets and these planets are shifted toward shorter periods.
More Planet Mass in Heavy Elements around Lower-Mass Stars
In a second study led by Gijs Mulders and Ilaria Pascucci we estimated the heavy element
budget that is stored in planets in the inner planetary systems as a function of stellar mass. The conversion of planet radius to planet mass was done through mass-radius relationships and we explored the possible range of such relationships. The surprising result of our study is that lower-mass stars not only have more planets (as we found in our earlier study), but there is more heavy element mass in those planets than in those around sun-like stars. This goes contrary to the expectation that low-mass stars have more planets because they fail to form larger ones. The trend we identify is also opposite to the trend found for dust mass in disks as a function of disk mass.