We conduct a statistical analysis of a combined sample of direct imaging data, totalling nearly 250 stars. The stars cover a wide range of ages and spectral types, and include five detections ($kappa$ And b, two $sim$60 M$_{rm J}$ brown dwarf companions in the Pleiades, PZ Tel B, and CD$-$35 2722B). For some analyses we add a currently unpublished set of SEEDS observations, including the detections GJ 504b and GJ 758B. We conduct a uniform, Bayesian analysis of all stellar ages using both membership in a kinematic moving group and activity/rotation age indicators. We then present a new statistical method for computing the likelihood of a substellar distribution function. By performing most of the integrals analytically, we achieve an enormous speedup over brute-force Monte Carlo. We use this method to place upper limits on the maximum semimajor axis of the distribution function derived from radial-velocity planets, finding model-dependent values of $sim$30--100 AU. Finally, we model the entire substellar sample, from massive brown dwarfs to a theoretically motivated cutoff at $sim$5 M$_{rm Jup}$, with a single power law distribution. We find that $p(M, a) propto M^{-0.65pm0.60} a^{-0.85pm0.39}$ (1$sigma$ errors) provides an adequate fit to our data, with 1.0--3.1% (68% confidence) of stars hosting 5--70 $M_{rm Jup}$ companions between 10 and 100 AU. This suggests that many of the directly imaged exoplanets known, including most (if not all) of the low-mass companions in our sample, formed by fragmentation in a cloud or disk, and represent the low-mass tail of the brown dwarfs.
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