A note on the Taylor series expansions for multivariate characteristics of classical risk processes.

The series expansion introduced by Frey and Schmidt (1996) [Taylor Series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 18, 1–12.] constitutes an original approach in approximating multivariate characteristics of classical ruin processes, specially ruin probabilities within finit time with certain surplus prior to ruin and severity of ruin. This approach can be considered alternative to inversion of Laplace transforms for particular claim size distributions [Gerber, H., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin 17(2), 151–163; Dufresne, F., Gerber, H., 1988a. The probability and severity of ruin for combinations of exponential claim amount distributions and their translations. Insurance: Mathematics and Economics 7, 75–80; Dufresne, F., Gerber, H., 1988b. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7, 193–199.] or discretization of the claim size and time [Dickson, C., 1989. Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics 8, 145–148; Dickson, C., Waters, H., 1992. The probability and severity of ruin in finit and infinit time. ASTIN Bulletin 22(2), 177–190; Dickson, C., 1993. On the distribution of the claim causing ruin. Insurance: Mathematics and Economics 12, 143–154.] applying the so-called Panjer’s recursive algorithm [Panjer, H.H., 1981. Recursive calculation of a family of compound distributions. ASTIN Bulletin 12, 22–26.]. We will prove that the recursive relation involved in the calculations of the the nth derivative with respect to – average number of claims in the time unit – of the multivariate finit time ruin probability (developed in the original paper by Frey and Schmidt (1996) can be simplified The cited simplificatio leads to a substantial reduction in the number of multiple integrals used in the calculations and makes the series expansion approach more appealing for practical implementationFinite time ruin probability; Surplus prior to ruin; Severity of ruin; Series expansion; Recursive methods;

An Advanced Mathematics Program for Middle School Teachers

The Conference Board of the Mathematical Sciences (CBMS), National Council of Teachers of Mathematics, and other organizations recommend twenty-one credits of mathematics coursework for prospective middle school teachers, beginning with a foundation based on mathematics for the elementary school curriculum, and followed by advanced courses directly addressing middle school mathematics. Three simultaneous factors—the emergence of the Interdisciplinary Liberal Studies Program at James Madison University, the release of CBMS guidelines, and a statewide focus on a critical shortage of qualified middle school teachers—provided an immediate audience for new upper-division courses built around the guidelines in probability/statistics, algebra, geometry, and calculus/analysis. We will discuss our experience with course planning and adaptation of other programs

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

We construct the universal type structure for conditional probability systems without any topological assumption, namely a type structure that is terminal, belief-complete, and non-redundant. In particular, in order to obtain the belief-completeness in a constructive way, we extend the work of Meier [An Infinitary Probability Logic for Type Spaces. Israel Journal of Mathematics, 192, 1-58] by proving strong soundness and strong completeness of an infinitary conditional probability logic with truthful and non-epistemic conditioning events.Comment: In Proceedings TARK 2017, arXiv:1707.0825

The mechanism behind probability

Changes within observable reality at the lowest level of reality seem to occur in accordance with the probability theory in mathematics. It is quite remarkable that nature itself has chosen the probability theory to arrange all the changes within the structure of the basic quantum fields. This rises a question about the distribution of properties in space and time

Four lectures on probabilistic methods for data science

Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional probability, focusing on the classical and matrix Bernstein's inequality and the uniform matrix deviation inequality. We illustrate these tools with applications for dimension reduction, network analysis, covariance estimation, matrix completion and sparse signal recovery. The lectures are geared towards beginning graduate students who have taken a rigorous course in probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of Data. Some typos, inaccuracies fixe

Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

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.Okazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.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.Andrzej Nędzusiak. s-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.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.Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.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.Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008, doi:10.2478/v10037-008-0017-z.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Bessel's inequality. Formalized Mathematics, 11(2):169-173, 2003.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

What makes mathematics lessons interesting in the middle school : student and teacher perceptions : a thesis presented in partial fulfilment of the requirements for the degree of Master of Educational Studies in Mathematics at Massey University

Some researchers have suggested that students in schools find mathematics classes boring, and that this attitude towards learning mathematics gets stronger as students grow older. Using reports of students and teachers, this study investigates how interest is used and developed in intermediate school mathematics classes. Five teachers and 101 Year 7 and 8 students from a single co-educational suburban state intermediate school participated in the study. One teacher and ten student focus group discussions to explore attitudes to and uses of interest in their mathematics classrooms were audio-taped. The results of these discussions were used to develop themes that formed the basis of separate student and staff questionnaires for all participants. Further data was obtained from a mathematics class journal kept by participants, and from individual interviews with all staff and seven randomly chosen students. The study showed that both teachers and students had similar ideas about what students found interesting, and revealed several aspects of classroom practices that heightened and/or developed interest in learning mathematics. The most notable of these were: using hands-on activities; teacher enthusiasm; group work and student progress. Mathematical content was rarely seen as interesting in itself, although probability, symmetry and transformations, geometry and problem solving were regarded as the most interesting sub-strands of the curriculum, while number, measurement and 'all of mathematics' garnered least support. Bookwork using textbooks or worksheets was usually considered boring, and activities such as external mathematics competitions and challenging or easy mathematics polarised student opinion. Interest has a complex and generally positive association with learning. Student reports suggest that two interest factors that have the potential to be used more effectively in mathematics lessons are teacher enthusiasm and group work. The catch phase of situational interest, the aspect of interest most frequently used, was rarely developed further. This study suggests that mathematics learning will benefit from further developing interest in mathematics classes by linking situational interest factors with mathematical content, student experiences and clarity about each student's progress. Teachers need professional development and resource support for this to happen