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I know that I can calculate the probability distributions of $x$ and $p$ from the Wigner quasiprobability distribution, and I can calculate the probability distributions of other operators by calculating their eigenstates, Wigner transforming them, and projecting them onto the Wigner distribution of interest.

Is there some way to use the Wigner distribution to calculate probability distributions of operators without finding the eigenstates in the operator representation of QM? For example, how would I go about finding the probability that the energy is within a certain range?

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If I understood you correctly, this is a very basic application of the Wigner function, see, for example, eq.11 of This summary by W. Case –  bechira May 15 '14 at 14:41
@bechira: That summary doesn't include getting the full probability distributions for the Weyl transform of an operator, just expectation values. –  Dan May 21 '14 at 2:34
gotcha, good question! –  bechira May 21 '14 at 2:57

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