DSpace Kolekcja: http://hdl.handle.net/11320/3543 2020-02-25T22:54:41Z Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables http://hdl.handle.net/11320/3578 Tytu&#322;: Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables Autorzy: Okazaki, Hiroyuki; Shidama, Yasunari Abstrakt: In this article we continue formalizing probability and randomness started in , where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables. 2010-01-01T00:00:00Z Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces http://hdl.handle.net/11320/3577 Tytu&#322;: Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces Autorzy: Inoué, Takao; Endou, Noboru; Shidama, Yasunari Abstrakt: In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to  and ). 2010-01-01T00:00:00Z Riemann Integral of Functions R into C http://hdl.handle.net/11320/3576 Tytu&#322;: Riemann Integral of Functions R into C Autorzy: Miyajima, Keiichi; Kato, Takahiro; Shidama, Yasunari Abstrakt: In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the . 2010-01-01T00:00:00Z Counting Derangements, Non Bijective Functions and the Birthday Problem http://hdl.handle.net/11320/3575 Tytu&#322;: Counting Derangements, Non Bijective Functions and the Birthday Problem Autorzy: Kaliszyk, Cezary Abstrakt: The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of . 2010-01-01T00:00:00Z