Geometrical Domain of Spin - 1 / 2 Probability Mass Function

The quantum analogue of the classical characteristic function for a spin 1/2 assembly is considered and the probability mass function of the random vector associated with the assembly is derived. It is seen that the positive regions of Wigner and Margenau-Hill quasi distributions for the three components of spin, correspond to a trivariate probability mass function. We identify the domain of these positive regions as an Octahedron inscribed in the Bloch sphere with its vertices on the surface of the sphere. It is in this domain that a quantum characteristic function characterizing the quasi distribution, admits a probability mass function in IR . It is also observed that the classical variates X1, X2, X3 corresponding to the 3 spin operators σ1, σ2, σ3 in the domain, are independent iff the Bloch vector lies on any one of the axes.