  We introduce several notions of random positive operator valued measures
(POVMs), and we prove that some of them are equivalent. We then study
statistical properties of the effect operators for the canonical examples,
obtaining limiting eigenvalue distributions with the help of free probability
theory. Similarly, we obtain the large system limit for several quantities of
interest in quantum information theory, such as the sharpness, the noise
content, and the probability range. Finally, we study different compatibility
criteria, and we compare them for generic POVMs.
