We show that the probability distribution of the Greenberger-Horne-Zeilinger quantum state (GHZ) under local action of independent von Neumann measurements follows a convex distribution of two distributions. The coefficients of the combination are related to the equatorial parts of the measurements, and the distributions associated with those coefficients are associated with the real parts of the measurements. One possible application of the result is that it allows one to split into two pieces the simulation of the GHZ state. Simulating, in worst-case or in average case, a quantum state like the GHZ state with random resources, shared or private, as well as with classical communication resources or even odd resources like nonlocal boxes is a very important in the theory of quantum communication complexity.