Exoplanets commonly occur and have been detected using a variety of techniques over the past thirty years. The newer direct imaging detection technique has already provided unique data and will continue to generate discoveries as instrumentation improves. Currently, only the brightest, self-luminous, and youngest planets have been imaged from the ground. Detecting and characterizing smaller, Earth-size planets will require dedicated space-based instrumentation. Given the high cost and complexity of space observatories, building confidence in a proposed instrument's capabilities prior to construction and deployment is vital. Predicting the performance of space-based exoplanet imagers depends on various assumptions made about the nature of the exoplanet population. Typically, these assumptions are extrapolations of the partially constrained distributions of planetary orbital and physical parameters derived from the currently known sample of exoplanets. From these extrapolated parameter distributions, distributions of derived parameters can be calculated and combined with an instrument's performance to yield metrics estimating the number of exoplanets an instrument will detect. In this dissertation, probability theory is used to derive analytical metrics of exoplanet direct imaging detection and fit planet occurrence rate density models to data from the literature. First, an analytical derivation is presented for single-visit completeness, the probability of detecting planets belonging to an assumed planet population for a given direct imaging instrument. This derivation is extended to determine the probability density functions of detected exoplanet population parameters. A depth-of-search metric is then derived which explicitly separates the effects of instrument performance from the assumptions on the planet population. Finally, planet occurrence rate density models fit to data from the literature as a function of planetary parameters and stellar effective temperature are presented. These metrics enhance our ability to predict science yields in the early stages of mission and instrument design while the occurrence rate density models are a step toward a more complete understanding of planet occurrence rates.