  We prove the existence for each Hilbert space of the two new quasi hidden variable (qHV) models, statistically noncontextual and context-invariant, reproducing all the von Neumann joint probabilities via nonnegative values of real-valued measures and all the quantum product expectations -- via the qHV (classical-like) average of the product of the corresponding random variables. In a context-invariant model, a quantum observable X can be represented by a variety of random variables satisfying the functional condition required in quantum foundations but each of these random variables equivalently models X under all joint von Neumann measurements, regardless of their contexts. The proved existence of this model negates the general opinion that, in terms of random variables, the Hilbert space description of all the joint von Neumann measurements for dimH>2 can be reproduced only contextually. The existence of a statistically noncontextual qHV model, in particular, implies that every N-partite quantum state admits a local quasi hidden variable (LqHV) model introduced in [Loubenets, J. Math. Phys. 53, 022201 (2012)]. The new results of the present paper point also to the generality of the quasi-classical probability model proposed in [Loubenets, J. Phys. A: Math. Theor. 45, 185306 (2012)]. 
Context-invariant quasi hidden variable (qHV) modelling of all joint von   Neumann measurements for an arbitrary Hilbert space