# Probability on Finite Set and Real-Valued Random Variables

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## Probability on Finite Set and Real-Valued Random Variables

In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.

[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. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.

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

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

[8] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996.

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

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

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

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

[13] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

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

[15] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.

[16] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.

[17] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.

[18] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.

[19] Grigory E. Ivanov. Definition of convex function and Jensen's inequality. Formalized Mathematics, 11(4):349-354, 2003.

[20] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.

[21] Jarosław Kotowicz and Yuji Sakai. Properties of partial functions from a domain to the set of real numbers. Formalized Mathematics, 3(2):279-288, 1992.

[22] Keiko Narita, Noboru Endou, and Yasunari Shidama. Integral of complex-valued measurable function. Formalized Mathematics, 16(4):319-324, 2008, doi:10.2478/v10037-008-0039-6.

[23] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990.

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

[25] Jan Popiołek. Introduction to probability. Formalized Mathematics, 1(4):755-760, 1990.

[26] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006, doi:10.2478/v10037-006-0018-8.

[27] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.

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

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

[30] 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

## (a computer assisted approach)

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

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