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Abstract
Large-scale surveys of transiting exoplanets such as \textit{Kepler} have revolutionized the study of exoplanet demographics. This dissertation focuses on the insights into planetary composition that can be gained by combining transit surveys with constraints on those planets' masses from radial velocity follow-up and transit timing variations. We first investigate second-order effects on the empirical planet mass-radius relation, finding the sample of \textit{Kepler} and other transiting planets from small surveys with mass measurements to be consistent with no dependence on host star mass. We then show how the joint mass-radius-period distribution of planets can be constrained using a mixture model to include several compositional subpopulations. We create a suite of models and employ model selection techniques to show that the inclusion of at least three subpopulations (planets with gaseous envelopes, evaporated rocky cores, and intrinsically rocky planets) is supported by the data. We find similar support for models that include or exclude photoevaporation, as well as models that include or exclude water worlds, highlighting the degeneracies inherent to the planet population in the mass-radius-period plane. We use our models to calculate $\eta_\oplus$, finding our estimate to be highly model dependent, obtaining a significantly lower $\eta_\oplus$ when photoevaporation is included in the model. Finally, we evolve a dense grid of planet evolution models that can enable future population studies to translate between the fundamental properties of planets (mass, envelope mass fraction, incident flux, age) and the observable plane (mass, radius, period). This thesis builds towards the robust characterization of the exoplanet composition distribution which will provide insights into competing theories of planet formation and the occurrence rate of habitable planets in the galaxy.