  We introduce probabilistic frames to study finite frames whose elements are
chosen at random. While finite tight frames generalize orthonormal bases by
allowing redundancy, independent, uniformly distributed points on the sphere
approximately form a finite unit norm tight frame (FUNTF). In the present
paper, we develop probabilistic versions of tight frames and FUNTFs to
significantly weaken the requirements on the random choice of points to obtain
an approximate finite tight frame. Namely, points can be chosen from any
probabilistic tight frame, they do not have to be identically distributed, nor
have unit norm. We also observe that classes of random matrices used in
compressed sensing are induced by probabilistic tight frames.
