Preface to pfeiffer applied probability  (Page 2/2)

Certain concepts and patterns have emerged from experience and intuition. The mathematical formulation (the mathematical model) which has most successfully captured these essential ideas is rootedin measure theory, and is known as the Kolmogorov model , after the brilliant Russian mathematician A.N. Kolmogorov (1903-1987).

One cannot prove that a model is correct . Only experience may show whether it is useful (and not incorrect). The usefulness of the Kolmogorov model is establishedby examining its structure and showing that patterns of uncertainty and likelihood in any practical situation can be represented adequately. Developments, such as in this course, have given ample evidence of such usefulness.

The most fruitful approach is characterized by an interplay of

• A formulation of the problem in precise terms of the model and careful mathematical analysis of the problem so formulated.
• A grasp of the problem based on experience and insight. This underlies both problem formulation and interpretation of analytical results of the model. Often such insight suggestsapproaches to the analytical solution process.

Matlab: a tool for learning

In this work, we make extensive use of MATLAB as an aid to analysis. I have tried to write the MATLAB programs in such a way that they constitute useful,ready-made tools for problem solving. Once the user understands the problems they are designed to solve, the solution strategies used, and the manner in which thesestrategies are implemented, the collection of programs should provide a useful resource.

However, my primary aim in exposition and illustration is to aid the learning process and to deepen insight into the structure of the problems considered and the strategies employed in their solution. Several features contribute to that end.

1. Application of machine methods of solution requires precise formulation. The data available and the fundamental assumptions must be organized in an appropriate fashion.The requisite discipline for such formulation often contributes to enhanced understanding of the problem.
2. The development of a MATLAB program for solution requires careful attention to possible solution strategies. One cannot instruct the machine without a clear grasp of what isto be done.
3. I give attention to the tasks performed by a program, with a general description of how MATLAB carries out the tasks. The reader is not required to trace out all theprogramming details. However, it is often the case that available MATLAB resources suggest alternative solution strategies. Hence, for those so inclined, attention to the details maybe fruitful. I have included, as a separate collection, the m-files written for this work. These may be used as patterns for extensions as well as programs in MATLAB for computations. Appendix A provides a directory of these m-files.
4. Some of the details in the MATLAB script are presentation details. These are refinements which are not essential to the solution of the problem. But they makethe programs more readily usable. And they provide illustrations of MATLAB techniques for those who may wish to write their own programs. I hope many will be inclined togo beyond this work, modifying current programs or writing new ones.

An invitation to experiment and explore

Because the programs provide considerable freedom from the burden of computation and the tyranny of tables (with their limited ranges and parameter values), standardproblems may be approached with a new spirit of experiment and discovery. When a program is selected (or written), it embodies one method of solution.There may be others which are readily implemented. The reader is invited, even urged, to explore! The user may experiment to whatever degree he or shefinds useful and interesting. The possibilities are endless.

Acknowledgments

After many years of teaching probability, I have long since lost track of all those authors and books which have contributed to the treatmentof probability in this work. I am aware of those contributions and am most eager to acknowledge my indebtedness, althoughnecessarily without specific attribution.

The power and utility of MATLAB must be attributed to to the long-time commitment of Cleve Moler, who made the package available in the public domain for several years. The appearanceof the professional versions, with extended power and improved documentation, led to further appreciation and utilization of its potential in applied probability.

The Mathworks continues to develop MATLAB and many powerful "tool boxes," and to provide leadership in many phases of modern computation. They have generously made available MATLAB 7to aid in checking for compatibility the programs written with earlier versions. I have not utilized the full potential of this version for developing professional quality user interfaces, since Ibelieve the simpler implementations used herein bring the student closer to the formulation and solution of the problems studied.

Connexions

Paul E. Pfeiffer

Rice University