Advanced Probability Appendix C

All professional education content courses leading to certification shall include teaching and assessment of the Wisconsin Content Standards in the content area.

In this column, list the Wisconsin Content Standards that are included in this course. The Standards for each content area are found in the Wisconsin Content Standards document.

In this column, indicate the nature of the performance assessments used in this course to evaluate student proficiency in each standard.

The structures within the discipline, the historical roots and evolving nature of mathematics, and the interaction between technology and the discipline.

Examples appear in class, on tests, and in assignments that show the use of technology to simulate Markov Chains and other random processes. The models studied in this course are classical and frame the basic questions that are at the heart of the development of modern probability theory. These models arise frequently in the mathematical formulation of problems encountered in real-world settings.

Facilitating the building of student conceptual and procedural understanding.

Markov chains and other basic random processes that are studied in this course have both well-defined procedural solutions and an underlying mathematical theory that lends itself to a conceptual understanding of these solutions.

Helping all students build understanding of the discipline including:

. Confidence in their abilities to utilize mathematical knowledge.

. Awareness of the usefulness of mathematics.

. The economic implications of fine mathematical preparation.

The problems on homework assignments and exams require students to integrate a rich background of material from Calculus and Discrete Mathematics. Many "real-world" problems can be formulated in terms of Markov Chains and other basic random processes that are studied in this course. The examples that we study from queuing theory and other topics in this course arise in the form of optimization problems that on a larger scale, such as is encountered in industry, are of considerable economic importance.

Exploring, conjecturing, examining and testing all aspects of problem solving.

A strong emphasis is placed on problem solving in this course. The use of computers to simulate random processes yield important opportunities to carry out the multiple stages of active problem solving.

Formulating and posing worthwhile mathematical tasks, solving problems using several strategies, evaluating results, generalizing solutions, using problem solving approaches effectively, and applying mathematical modeling to real-world situations.

The determination of steady-states solutions for Markov Chains can be approached in several directions - recurrence relations, matrix operations. The determination of recurrence times and the times needed to reach certain ergodic states can be developed as generalizations of geometric random variables. The results so obtained can be verified through the simulation by a computer of the accompanying random processes.

Making convincing mathematical arguments, framing mathematical questions and conjectures, formulating counter-examples, constructing and evaluating arguments, and using intuitive, informal exploration and formal proof.

Students are required to reproduce the proof of certain classical theorems, i.e.,, those that characterize certain process as being Poisson processes based on their being time invariant, etc.

Expressing ideas orally, in writing, and visually-, using mathematical language, notation, and symbolism; translating mathematical ideas between and among contexts.

Students regular submit homework assignments and take exams; in-class participation is strongly encouraged.

Connecting the concepts and procedures of mathematics, drawing connections between mathematical strands, between mathematics and other disciplines, and with daily life.

The mathematical models encountered in this course have a celebrated place in the study of gambling and the growth of populations.

Selecting appropriate representations to facilitate mathematical problem solving and translating between and among representations to explicate problem-solving situations.

Where possible, the models encountered in Advanced Probability are translated into terms that allow them to be solved by using calculus or discrete mathematics.

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

In class work and out-of-class assignments lead to the development of problem solving and communication on the part of the student. The topics in advanced probability introduce problems whose solutions introduce the mathematical concepts from any one of a number of other mathematics courses in the curriculum: discrete mathematics, calculus, linear algebra, mathematical programming, differential equations, and abstract algebra.

Number operations and relationships from both abstract and concrete perspectives identifying real world applications, and representing and connecting mathematical concepts and procedures including:

. Number sense.

. Set theory.

. Number and operation.

. Composition and decomposition of numbers, including place value, primes, factors, multiples, inverses, and the extension of these concepts throughout mathematics.

. Number systems through the real numbers, their properties and relations.

. Computational procedures.

. Proportional reasoning.

. Number theory.


Mathematical concepts and procedures, and the connections among them for teaching upper level number operations and relationships including:

. Advanced counting procedures, including union and intersection of sets, and parenthetical operations.

. Algebraic and transcendental numbers.

. The complex number system, including polar coordinates.

. Approximation techniques as a basis for numerical integration, fractals, and numerical-based proofs.

. Situations in which numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social) can be created and critically evaluated.

. Opportunities in which acceptable limits of error can be assessed (e.g., evaluating strategies, testing the reasonableness of results, and using technology to carry out computations).

In Advanced Probability, advanced counting techniques are used in enumerating the number of ways in which a given event can occur. These methods of enumeration are then reformulated in terms of linear algebra and matrix operations. The answers to queries about the random processes that we study have values that can be intuited from a basic consideration of the qualitative properties of the process. This relationship allows students the opportunity to assess the reasonableness of their results. The underlying use of linear algebra, matrix methods, and differential equation in Advanced Probability allows for the use of technology to carry out the computations therein.

Geometry and measurement from both abstract and concrete perspectives and to identify real world applications, and mathematical concepts, procedures and connections among them including:

. Formal and informal argument.

. Names, properties, and relationships of two- and three-dimensional shapes.

. Spatial sense.

. Spatial reasoning and the use of geometric models to represent, visualize, and solve problems.

. Transformations and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships.

. Coordinate geometry systems including relations between coordinate and synthetic geometry, and generalizing geometric principles from a two-dimensional system to a three-dimensional system.

. Concepts of measurement, including measurable attributes, standard and non-standard units, precision and accuracy, and use of appropriate tools.

. The structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems. Measurement including length, area, volume, size of angles, weight and mass, time, temperature, and money.

. Measuring, estimating, and using measurement to describe and compare geometric phenomena.

. Indirect measurement and its uses, including developing formulas and procedures for determining measure to solve problems.

The study of two-dimension random walk introduces the technique of identifying the arrival times for certain symmetrically placed sites as being equal; this use of symmetry allows one to reduce the size of the state space that must be employed in the construction of equations for the arrival times of the random walk on the sites of a lattice.

Mathematical concepts, procedures, and the connections among them for teaching upper level geometry and measurement including:

. Systems of geometry, including Euclidean, non-Euclidean, coordinate, transformational, and projective geometry.

. Transformations, coordinates, and vectors and their use in problem solving. Three-dimensional geometry and its generalization to other dimensions. Topology, including topological properties and transformations.

. Opportunities to present convincing arguments by means of demonstration, informal proof, counter-examples, or other logical means to show the truth of statements and/or generalizations.


Statistics and probability from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections between them including:

. Use of data to explore real-world issues.

. The process of investigation including formulation of a problem, designing a data collection plan, and collecting, recording, and organizing data.

. Data representation through graphs, tables, and summary statistics to describe data distributions, central tendency, and variance.

. Analysis and interpretation of data.

. Randomness, sampling, and inference.

. Probability as a way to describe chances or risk in simple and compound events.

. Outcome prediction based on experimentation or theoretical probabilities.


Mathematical concepts, procedures, and the connections among them for teaching upper level statistics and probability including:

. Use of the random variable in the generation and interpretation of probability distributions.

. Descriptive and inferential statistics, measures of disbursement, including validity and reliability, and correlation.

. Probability theory and its link to inferential statistics.

. Discrete and continuous probability distributions as bases for inference.

. Situations in which students can analyze, evaluate, and critique the methods and conclusions of statistical experiments reported in journals, magazines, news media, advertising, etc.

The use of random variables, such as the state of a Markov chain, as well as the arrival and recurrence times for the same, are ubiquitous in Advanced Probability.

Functions, algebra, and basic concepts underlying calculus from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections among them including:

. Patterns.

. Functions as used to describe relations and to model real world situations.

. Representations of situations that involve variable quantities with expressions, equations and inequalities and that include algebraic and geometric relationships.

. Multiple representations of relations, the strengths and limitations of each representation, and conversion from one representation to another.

. Attributes of polynomial, rational, trigonometric, algebraic, and exponential functions.

. Operations on expressions and solution of equations, systems of equations and inequalities using concrete, informal, and formal methods.

. Underlying concepts of calculus, including rate of change, limits, and approximations for irregular areas.

Recurrence times, first arrival times, and occupation probabilities are frequently calculated by solving systems of equations.

Mathematical concepts, procedures, and the connections among them for teaching upper level functions, algebra, and concepts of calculus including:

. Concepts of calculus, including limits (epsilon-delta) and tangents, derivatives, integrals, and sequences and series.

. Modeling to solve problems.

. Calculus techniques including finding limits, derivatives, integrals, and using special rules.

. Calculus applications including modeling, optimization, velocity and acceleration, area, volume, and center of mass.

. Numerical and approximation techniques including Simpson's rule, trapezoidal rule, Newton's Approximation, and linearization.

. Multivariate calculus.

. Differential equations.

By ergodic theory, the equilibrium state of a Markov Chain is the limit of the chains behavior as the time parameter tends to infinity.

Basic problems in gambling theory and population dynamics can be formulated in terms of Markov chains and other random processes.

The communicating classes of a Markov Chain form an equivalence class on the state space of the chain.

Discrete processes from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections among them including:

. Counting techniques.

. Representation and analysis of discrete mathematics problems using sequences, graph theory, arrays, and networks.

. Iteration and recursion.

Counting techniques are used to identify the number of ways in which an event can occur; these enumerations are basic to the calculation of certain probabilities that arise in a natural way in the study of Markov Chains and other models.

Markov chains can be represented in a natural way in terms of weighted, directed graphs by drawing edges between communicating states and employing the corresponding transition probability between these states as the weight.

Recurrence and arrival times can be determined by formulating and solving certain recurrence relations.

Mathematical concepts, procedures, and the connections among them for teaching upper level discrete mathematics including:

. Topics, including symbolic logic, induction, linear programming, and finite graphs.

. Matrices as a mathematical system, and matrices and matrix operations as tools for recording information and for solving problems.

. Developing and analyzing algorithms.


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Department of Mathematics
Laurel Langford, Department Chair
207 North Hall
410 S. 3rd Street, River Falls, WI 54022-5001


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