Characterization of the joint (among wavebands) probability density function (PDF) of hyperspectral imaging (HSI) data is crucial for several applications, including the design of constant false alarm rate (CFAR) detectors and statistical classifiers. HSI data are vector (or equivalently multivariate) data in a vector space with dimension equal to the number of spectral bands. As a result, the scalar statistics utilized by many detection and classification algorithms depend upon the joint pdf of the data and the vector-to-scalar mapping defining the specific algorithm. For reasons of analytical tractability, the multivariate Gaussian assumption has dominated the development and evaluation of algorithms for detection and classification in HSI data, although it is widely recognized that it does not always provide an accurate model for the data. The purpose of this paper is to provide a detailed investigation of the joint and marginal distributional properties of HSI data. To this end, we assess how well the multivariate Gaussian pdf describes HSI data using univariate techniques for evaluating marginal normality, and techniques that use unidimensional views (projections) of multivariate data. We show that the class of elliptically contoured distributions, which includes the multivariate normal distribution as a special case, provides a better characterization of the data. Finally, it is demonstrated that the class of univariate stable random variables provides a better model for the heavy-tailed output distribution of the well known matched filter target detection algorithm.