  We propose a general framework of the quantum/quasi-classical transformations
by introducing the concept of quasi-joint-spectral distribution (QJSD).
Specifically, we show that the QJSDs uniquely yield various pairs of
quantum/quasi-classical transformations, including the Wigner-Weyl transform.
We also discuss the statistical behaviour of combinations of generally
non-commutin quantum observables by introducing the concept of quantum
correlations and conditional expectations defined analogously to the classical
counterpart. Based on these, Aharonov's weak value is given a statistical
interpretation as one realisation of the quantum conditional expectations
furnished in our formalism.
