Abstract
The notion of a hyper-random event is introduced. To describe such events, conditional probabilities and probability bounds are used. The properties of these parameters are presented. The concept of a scalar hyper-random variable is introduced. Here we use conditional distribution functions (providing an exhaustive description), bounds of the distribution function, moments of the distribution function, and bounds of these moments. The properties of these characteristics and parameters are presented. The notion of a hyper-random vector variable is introduced. Methods used to describe hyper-random scalar variables are extended to the case of hyper-random vector variables. Properties of the characteristics and parameters of hyper-random vector variables are given.
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Notes
- 1.
Hereafter, a tilde under a letter is used to indicate that the given object is or may be many-valued.
- 2.
In the general case, the formulas
\(P_{I} \left( {\mathop \bigcup\limits_{m = 1}^{\infty } {A_{m} } } \right) = \mathop {\lim }\limits_{M \to \infty } P_{I} \left( {A_{M} } \right)\), \(P_{S} \left( {\mathop \bigcap\limits_{m = 1}^{\infty } {A_{m} } } \right) = \mathop {\lim }\limits_{M \to \infty } P_{S} \left( {A_{M} } \right)\)
for \(A_{m} \subset A_{m + 1}\) and \(A_{m + 1} \subset A_{m}\) (\(m \ge 1\)) are not correct. The author would like to thank Professor V.N. Tutubalin for drawing his attention to this fact.
- 3.
It is assumed here and below that all the above distribution functions are continuous or piecewise continuous.
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Gorban, I.I. (2017). Hyper-random Events and Variables. In: The Statistical Stability Phenomenon. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-43585-5_9
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DOI: https://doi.org/10.1007/978-3-319-43585-5_9
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