Random Variables and Product of Probability Spaces

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We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of countably infinite probability spaces.

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[5] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.

[6] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.

[7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

[8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[12] Peter Jaeger. Elementary introduction to stochastic finance in discrete time. FormalizedMathematics, 20(1):1-5, 2012. doi:10.2478/v10037-012-0001-5.

[13] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.

[14] Hiroyuki Okazaki. Probability on finite and discrete set and uniform distribution. FormalizedMathematics, 17(2):173-178, 2009. doi:10.2478/v10037-009-0020-z.

[15] Hiroyuki Okazaki and Yasunari Shidama. Probability on finite set and real-valued random variables. Formalized Mathematics, 17(2):129-136, 2009. doi:10.2478/v10037-009-0014-x.

[16] Hiroyuki Okazaki and Yasunari Shidama. Probability measure on discrete spaces and algebra of real-valued random variables. Formalized Mathematics, 18(4):213-217, 2010. doi:10.2478/v10037-010-0026-6.

[17] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

[18] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

[21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

[22] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. The relevance of measure and probability, and definition of completeness of probability. Formalized Mathematics, 14 (4):225-229, 2006. doi:10.2478/v10037-006-0026-8.

Formalized Mathematics

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SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237

Target Group

researchers in the fields of formal methods and computer-checked mathematics


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