  Noncommuting observables cannot be simultaneously measured, however, under
local hidden variable models, they must simultaneously hold premeasurement
values, implying the existence of a joint probability distribution. We study
the joint distributions of noncommuting observables on qubits, with possible
criteria of positivity and the Fr\'echet bounds limiting the joint
probabilities, concluding that the latter may be negative. We use
symmetrization, justified heuristically and then more carefully via the Moyal
characteristic function, to find the quantum operator corresponding to the
product of noncommuting observables. This is then used to construct Quasi-Bell
inequalities, Bell inequalities containing products of noncommuting
observables, on two qubits. These inequalities place limits on local hidden
variable models that define joint probabilities for noncommuting observables.
We find Quasi-Bell inequalities have a quantum to classical violation as high
as $\frac{3}{2}$, higher than conventional Bell inequalities. The result
demonstrates the theoretical importance of noncommutativity in the nonlocality
of quantum mechanics, and provides an insightful generalization of Bell
inequalities.
