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We prove the existence for each Hilbert space of the two new quasi hiddenvariable (qHV) models, statistically noncontextual and context-invariant,reproducing all the von Neumann joint probabilities via nonnegative values ofreal-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 avariety of random variables satisfying the functional condition required inquantum foundations but each of these random variables equivalently models Xunder all joint von Neumann measurements, regardless of their contexts. Theproved existence of this model negates the general opinion that, in terms ofrandom variables, the Hilbert space description of all the joint von Neumannmeasurements for dimH>2 can be reproduced only contextually. The existence of astatistically noncontextual qHV model, in particular, implies that everyN-partite quantum state admits a local quasi hidden variable (LqHV) modelintroduced in [Loubenets, J. Math. Phys. 53, 022201 (2012)]. The new results ofthe present paper point also to the generality of the quasi-classicalprobability 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