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This thesis is devoted to Bernoulli Stochastics, which was initiated by Jakob Bernoulli more than 300 years ago by his master piece 'Ars conjectandi', which can be translated as 'Science of Prediction'. Thus, Jakob Bernoulli's Stochastics focus on prediction in contrast to the later emerging disciplines probability theory, statistics and mathematical statistics. Only recently Jakob Bernoulli's focus was taken up von Collani, who developed a unified theory of uncertainty aiming at making reliable and accurate predictions. In this thesis, teaching material as well as a virtual classroom are developed for fostering ideas and techniques initiated by Jakob Bernoulli and elaborated by Elart von Collani. The thesis is part of an extensively construed project called 'Stochastikon' aiming at introducing Bernoulli Stochastics as a unified science of prediction and measurement under uncertainty. This ambitious aim shall be reached by the development of an internet-based comprehensive system offering the science of Bernoulli Stochastics on any level of application. So far it is planned that the 'Stochastikon' system (http://www.stochastikon.com/) will consist of five subsystems. Two of them are developed and introduced in this thesis. The first one is the e-learning programme 'Stochastikon Magister' and the second one 'Stochastikon Graphics' that provides the entire Stochastikon system with graphical illustrations. E-learning is the outcome of merging education and internet techniques. E-learning is characterized by the facts that teaching and learning are independent of place and time and of the availability of specially trained teachers. Knowledge offering as well as knowledge transferring are realized by using modern information technologies. Nowadays more and more e-learning environments are based on the internet as the primary tool for communication and presentation. E-learning presentation tools are for instance text-files, pictures, graphics, audio and videos, which can be networked with each other. There could be no limit as to the access to teaching contents. Moreover, the students can adapt the speed of learning to their individual abilities. E-learning is particularly appropriate for newly arising scientific and technical disciplines, which generally cannot be presented by traditional learning methods sufficiently well, because neither trained teachers nor textbooks are available. The first part of this dissertation introduces the state of the art of e-learning in statistics, since statistics and Bernoulli Stochastics are both based on probability theory and exhibit many similar features. Since Stochastikon Magister is the first e-learning programme for Bernoulli Stochastics, the educational statistics systems is selected for the purpose of comparison and evaluation. This makes sense as both disciplines are an attempt to handle uncertainty and use methods that often can be directly compared. The second part of this dissertation is devoted to Bernoulli Stochastics. This part aims at outlining the content of two courses, which have been developed for the anticipated e-learning programme Stochastikon Magister in order to show the difficulties in teaching, understanding and applying Bernoulli Stochastics. The third part discusses the realization of the e-learning programme Stochastikon Magister, its design and implementation, which aims at offering a systematic learning of principles and techniques developed in Bernoulli Stochastics. The resulting e-learning programme differs from the commonly developed e-learning programmes as it is an attempt to provide a virtual classroom that simulates all the functions of real classroom teaching. This is in general not necessary, since most of the e-learning programmes aim at supporting existing classroom teaching. The forth part presents two empirical evaluations of Stochastikon Magister. The evaluations are performed by means of comparisons between traditional classroom learning in statistics and e-learning of Bernoulli Stochastics. The aim is to assess the usability and learnability of Stochastikon Magister. Finally, the fifth part of this dissertation is added as an appendix. It refers to Stochastikon Graphics, the fifth component of the entire Stochastikon system. Stochastikon Graphics provides the other components with graphical representations of concepts, procedures and results obtained or used in the framework of Bernoulli Stochastics. The primary aim of this thesis is the development of an appropriate software for the anticipated e-learning environment meant for Bernoulli Stochastics, while the preparation of the necessary teaching material constitutes only a secondary aim used for demonstrating the functionality of the e-learning platform and the scientific novelty of Bernoulli Stochastics. To this end, a first version of two teaching courses are developed, implemented and offered on-line in order to collect practical experiences. The two courses, which were developed as part of this projects are submitted as a supplement to this dissertation. For the time being the first experience with the e-learning programme Stochastikon Magister has been made. Students of different faculties of the University of Würzburg, as well as researchers and engineers, who are involved in the Stochastikon project have obtained access to Stochastikon Magister via internet. They have registered for Stochastikon Magister and participated in the course programme. This thesis reports on two assessments of these first experiences and the results will lead to further improvements with respect to content and organization of Stochastikon Magister.
Das stochastische Denken, die Bernoullische Stochastik und dessen informationstechnologische Umsetzung, namens Stochastikon stellen die Grundlage für das Verständnis und die erfolgreiche Nutzung einer stochastischen Wissenschaft dar. Im Rahmen dieser Arbeit erfolgt eine Klärung des Begriffs des stochastischen Denkens, eine anschauliche Darstellung der von Elart von Collani entwickelten Bernoullischen Stochastik und eine Beschreibung von Stochastikon. Dabei werden sowohl das Gesamtkonzept von Stochastikon, sowie die Ziele, Aufgaben und die Realisierung der beiden Teilsysteme namens Mentor und Encyclopedia vorgestellt. Das stochastische Denken erlaubt eine realitätsnahe Sichtweise der Dinge, d.h. eine Sichtweise, die mit den menschlichen Beobachtungen und Erfahrungen im Einklang steht und somit die Unsicherheit über zukünftige Entwicklungen berücksichtigt. Der in diesem Kontext verwendete Begriff der Unsicherheit bezieht sich ausschließlich auf zukünftige Entwicklungen und äußert sich in Variabilität. Quellen der Unsicherheit sind einerseits die menschliche Ignoranz und andererseits der Zufall. Unter Ignoranz wird hierbei die Unwissenheit des Menschen über die unbekannten, aber feststehenden Fakten verstanden, die die Anfangsbedingungen der zukünftigen Entwicklung repräsentieren. Die Bernoullische Stochastik liefert ein Regelwerk und ermöglicht die Entwicklung eines quantitativen Modells zur Beschreibung der Unsicherheit und expliziter Einbeziehung der beiden Quellen Ignoranz und Zufall. Das Modell trägt den Namen Bernoulli-Raum und bildet die Grundlage für die Herleitung quantitativer Verfahren, um zuverlässige und genaue Aussagen sowohl über die nicht-existente zufällige Zukunft (Vorhersageverfahren), als auch über die unbekannte feststehende Vergangenheit (Messverfahren). Das Softwaresystem Stochastikon implementiert die Bernoullische Stochastik in Form einer Reihe autarker, miteinander kommunizierender Teilsysteme. Ziel des Teilsystems Encyclopedia ist die Bereitstellung und Bewertung stochastischen Wissens. Das Teilsystem Mentor dient der Unterstützung des Anwenders bei der Problemlösungsfindung durch Identifikation eines richtigen Modells bzw. eines korrekten Bernoulli-Raums. Der Lösungsfindungsprozess selber enthält keinerlei Unsicherheit. Die ganze Unsicherheit steckt in der Lösung, d.h. im Bernoulli-Raum, der explizit die vorhandene Unwissenheit (Ignoranz) und den vorliegenden Zufall abdeckend enthält.