The uncertainty principle in signal analysis

We give a simple derivation for the covariance of time and frequency. The simplicity remains when applied to arbitrary variables. We show that the covariance enters into the uncertainty principle in a fundamental way. Also we present a seeming paradox: the uncertainty principle depends only on the marginals, but the marginals carry no information about covariance. Then how is it possible that the uncertainty principle involves the covariance? The resolution of the paradox clarifies a number of issues regarding the existence of well behaved manifestly positive joint distributions of time and frequency.<>