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Symplectic and optical joint probability representations of quantum mechanicsare considered, in which the functions describing the states are theprobability distributions with all random arguments (except the argument oftime ). The general formalism of quantizers and dequantizers determining thestar product quantization scheme in these representations is given. Taking theGaussian functions as the distributions of the tomographic parameters thecorrespondence rules for most interesting physical operators are found and theexpressions of the dual symbols of operators in the form of singular andregular generalized functions are derived. Evolution equations and stationarystates equations for symplectic and optical joint probability distributions areobtained.

Observables, evolution equation,and stationary states equation in the joint probability representation of quantum mechanics