We consider  discrete distributions of random variables, defined by various two-symbol systems of encoding of real numbers (with zero and non-zero redundancy, with one and two bases, in particular with different sings), and study structural, topological, metric, and structurally fractal properties  their point spectra. The general criterion for random variable with independent digits of two-symbol representation to have discrete distribution (analog of the P. L'\ evi theorem for sum of random series with discretely distributed terms) is proved and properties of its spectrum are described.  In the paper we study discrete distributions of values of functions of the Cantor type of a random continuously distributed argument.

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