Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation . We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.
In this article we continue formalizing probability and randomness started in [13], 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.
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.
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.
Článek se věnuje známé matematické úloze, která nese název problém čtyř bodů. Její řešení není jednoznačné, liší se podle volby pravděpodobnostního rozdělení bodů. Zde uvádíme řešení při volbě rovnoměrného rozdělení na množinách různých tvarů a porovnáváme jednotlivé výsledky.