Probable Course Research Articles

Bold Play Is Best: A Simple Proof

Introduction In the classical ruin problem, a gambler with initial capital i dollars plays against an adversary with initial capital a i dollars. Here i and a are positive integers, i < a. The gambler wins a dollar with probability p and loses a dollar with probability q = 1 p; the game is repeated until one of the players goes broke. (Part of the analysis of the problem shows the game cannot go on forever.) Another interpretation of this model is that of a gambler playing a game in a casino (the adversary) where the gambler plays until she goes broke or until she wins a fixed predetermined amount, at which time she quits. Given a total fixed capital a, we are interested in the probability qi that the gambler starting with initial capital i, 1 < i < a 1, is ruined. This problem is often discussed in a first course in probability and introduces the student to the ideas of random walks and Markov chains. Our main reference is the classic book of Feller [3, chapter 14]; see also [1], [4], and [5]. We will regard the sequence of the gambler's fortune after each play as a random walk in the initerval [0, a], with absorbing barriers at 0 and a. The probability of ruin is the probability of hitting 0 before hitting a. Using a difference equation approach and some algebra, the following solution is derived in the case p = 1/2 (see, e.g., [3]):

Understanding the Nature of Errors in Probability Problem-Solving

This study provides an investigation of relationships among different types of errors occurring during probability problem-solving. Fifty non-mathematically sophisticated graduate student subjects enrolled in an introductory probability and statistics course were asked to solve a set of probability problems, and their attempts at solution were analyzed for presence and type of errors. The errors contained within these solutions were categorized according to a coding scheme which identifies 110 specific kinds of errors in four categories: text comprehension errors, conceptual errors, procedural errors, and arithmetic/computation errors. Relationships among types of errors included in each category were investigated using hierarchical clustering via additive trees. Implications of these relationships for the teaching and learning of probability problem-solving are discussed.

Classroom notes A physical interpretation of counting formulas and their comparison with each other

In afirst course in statistics or probability the student encounters problems of calculation of probabilities in finite sample spaces. Generally, this requires learning some basic counting formulas, and it is at this point where many students become confused. Frequently, confusion results from absence of a systematic presentation of four basic counting formulas.

On the Information Economics Approach to the Generalized Game Show Problem

The game show problem has received considerable attention since it appeared in Marilyn vos Savant's column in the Parade Magazine. I consider in this article the player's optimal decision strategy and the probability of winning the prize in a generalized version of the game show problem. By means of the information economics approach, we can easily (1) represent the host's various strategies as a simple matrix form; (2) extend the problem to more-than-three-door cases; (3) incorporate the player's prior information; and (4) consider the problem in which the player's choice behavior depends not on the probability of winning the prize, but on the expected utility in the generalized game show problem. The intricacies of this wonderfully confusing little problem make it an excellent tool in teaching probability and statistics courses.

Academic Emergency Medicine's Future

Emergency medicine (EM) will change over the next 20 years more than any other specialty. Its proximity to and interrelationships with the community, nearly all other clinicians (physicians and nonphysicians), and scientific/technologic developments guarantee this. While emergency physicians (EPs) will continue to treat both emergent and nonemergent patients, over the next decades our interventions, methods, and place in the medical care system will probably become unrecognizable from the EM we now practice and deliver. This paper, developed by the Society for Academic Emergency Medicine (SAEM) Task Force on Academic Emergency Medicine's Future, was designed to promote discussions about and actions to optimize our specialty's future. After briefly discussing the importance of futures planning, it suggests "best-case," "worst-case," and most probable future courses for academic EM over the next decades. The authors predict that EPs will practice a much more technologic and accurate form of medicine, with diagnostic, patient, reference, and consultant information rapidly available to them. They will be at the center of an extensive consultation network stemming from major medical centers and the purveyors of a sophisticated home health system, very similar to or even more advanced than what is now delivered on hospital wards. The key to planning for our specialty is for EM organizations, academic centers, and individuals to act now to optimize our possible future.

Radiologic staging of gastrointestinal cancer.

Pre-operative staging should define the probable course of a patient's disease, separate the resectable from the unresectable patients, and identify the patients who are candidates for induction therapy. Pre-operative staging must be well tolerated and should provide new or important information that will affect the proposed treatment plan. For some diseases such as colon cancer, pre-operative staging is largely unnecessary, but it can be of great value for other tumors such as pancreatic cancer. Currently, many imaging techniques are available to evaluate gastrointestinal cancers, and each provides information necessary for directing treatment. Although no individual technique can stage patients with complete accuracy, combinations of the various imaging techniques can be used to increase accuracy and promote appropriate decisions about an individual's treatment options. The use of current imaging techniques for staging primary lesions, regional spread, and the intra-abdominal metastatic spread of the most common gastrointestinal malignancies are reviewed.

A First Course in Probability

A First Course in Probability

A First Course in Probability

A First Course in Probability

The Case for Chaos

Chaos theory and fractal geometry have begun to appear as separate units in the mathematics curriculum of many high schools. Many more, however, have found these units appealing but impossible to squeeze into an already packed program. Yet a number of students at these same schools complete a precalculus sequence before the end of their senior year and find themselves, for one reason or another, not ready to go right on to the calculus. Typically, schools offer such students semester courses in probability, statistics, discrete mathematics or computer programming. In this article, I make a case for expanding this offering to include a semester of chaos theory that pulls together all those appealing units that may be scattered through the curriculum. After a brief overview of chaos theory and fractal geometry, I outline eight units that could constitute such a course and include a list of useful resources. For a number of years, I have taught a course based on this outline to precalculus students whose enthusiastic response is the motivation for this article.

Risk—A Motivating Theme for an Introductory Statistics Course

This article describes an idea for motivating students in an introductory probability and statistics course. The motivating theme is risk, and the process begins with a first-day-of-class questionnaire that samples attitudes of students toward risk and involves them in analysis of events and decisions from their daily lives. Questionnaire responses serve as a context for the instructor to develop the technical concepts of probability and statistics. Moreover, the questionnaire provides a way to increase substantially student motivation and involvement in the course.

STUDENT PROJECTS MAKE STATISTICS COME ALIVE FOR MATHEMATICS MAJORS

ABSTRACT This article suggests an approach in courses for mathematics majors and minors similar to the approach becoming popular in applied courses - using student projects. A natural place to introduce modern applications is in upper division, calculus-based probability and statistics courses for mathematics majors and minors. To provide students with an opportunity to design and study a real problem, students can be required to complete a statistical analysis of a problem in a semester-long project. The strategy and implementation is described in the article. 1 St. Joseph's University is a private, church affiliated university with approximately 3500 full-time students.

Analytical laws of mortality in actuarial mathematics

The choice of an appropriate statistical model for a real‐life application is often difficult. Standard examples given in a traditional calculus‐based course in probability and statistics do not provide students with a taste of the challenge usually associated with finding a satisfactory theoretical probability distribution in certain important applications. In addition, the assessment of the goodness of fit in practical terms is not usually discussed. To illustrate the difficulties of modelling, analytical laws of mortality are proposed using probability functions that are commonly discussed in a standard probability and statistics text intended for undergraduates in mathematics and statistics. An assessment is made of the resulting quality of approximations obtained for the valuation of premiums regarding a term life insurance policy.

An Intermediate Course in Probability.

An Intermediate Course in Probability.

Teacher’s Corner: Using Exams for Teaching Concepts in Probability and Statistics

We present several classroom demonstrations that have sparked student involvement in our introductory undergraduate courses in probability and statistics. The demonstrations involve both experimentation using exams and statistical analysis and adjustment of exam scores.

INTEGRATING UNCERTAINTY CONSIDERATIONS IN LEARNING ENGINEERING ECONOMY

The traditional approach to learning engineering economy in undergraduate curricula typically focuses on solving problems in a deterministic manner. Students generally have little exposure to the uncertain and stochastic nature of, as examples, project cash flows and interest rates. Unfortunately, this traditional approach does not provide students with the skills to deal with real world situations, which inherently involve uncertainty and thereby, risk. In general, engineering economy texts for undergraduate students deal with uncertainty and risk only in brief chapters, usually at the end of the book. The uncertain environment is introduced as a special case, rather than as the norm. In this paper, we propose an approach to learning engineering economy that is integrated with an understanding of uncertainty; in fact, it considers the deterministic case as a special case. The availability of computers today facilitates introducing this uncertainty approach. Computer spreadsheets and software can provide students with the ability to analyze entire probability distributions. Such a junior or senior level engineering economy course would build on an earlier course in probability and statistics, as well as courses in design and computing. A plausible new engineering economy text that incorporates stochastic considerations is outlined herein.

Ultrastructure and phylogeny of the spermatozoid ofChara vulgaris (Charophyceae)

At maturity, spermatozoids of the green algaChara vulgaris are biflagellated, contain little cytoplasm, and coil for approximately 2 1/2 gyres within the mother cell wall. The anterior of the cell contains an ovoid headpiece anchoring two slightly staggered basal bodies that are positioned above and directly in front of approximately 30 linearly arranged mitochondria. An elongated stellate pattern occupies the transition zone between the BBs and axonemes. Flagella emerge from the cell just in front of the nucleus and encircle the full length of the spermatozoid. The spline comprises a maximum of 38 microtubules surrounding the anterior mitochondria and gradually decreases posteriorly to a minimum of 11. The dense nucleus is narrow, cylindrical, and occupies the central revolution of the cell. Six starch-laden plastids and associated mitochondria are linearly arranged at the cell posterior. Phylogenetic analyses of charalean taxa and archegoniates based on spermatogenesis strongly support the orderCharales, withNitella as the sister group toChara. Diagnostic features ofChara spermatozoids include absence of a lamellar strip and axonemes embedded in the cell for almost the entire length of the anterior mitochondria. Potential relationships amongCharales, Coleochaetales and archegoniates are evaluated in regards to the probable course of evolution of streamlined biflagellated gametes.

CLASSICAL PROBABILITY APPLICATION: A DISCRETE DYNAMICAL SYSTEM DEFINED BY A MARKOV CHAIN

ABSTRACT Each student at the United States Military Academy takes a four course sequence in core mathematics. As the final core mathematics capstone course, probability and statistics (P&S) not only includes the standard material of such a course, but also asks the students to reflect upon previous core mathematics topics. Since our probability course covers continuous univariate and multivariate distributions, calculus (core courses 2 and 3) already assumes a significant role. It was our desire that the students utilize their modeling skills gained in the first core course, Discrete Dynamical Systems (DDS). This led to a P&S project that requires DDS to find the steady-state probabilities of a Markov Chain. These steady-state probabilities are then used in a larger Bayes' Theorem and conditional probability scenario.

Book Reviews

Abstract Abstract Statistical Analysis of Nonnormal Data J. V. Deshpande, A. P. Gore, and A. Shanubhogue. New York: Wiley, 1995. viii + 240 pp. $40.95. Reviewed by Subha Chakraborti University of Alabama-Tuscaloosa Straightforward Statistics For the Behavioral Sciences James D. Evans. Pacific Grove, CA: Brooks/Cole, 1996. xxii + 600 pp. $63.95. Reviewed by Karl L. Wuensch East Carolina University Growth Curves Anant M. Kshirsagar and William Boyce Smith. New York: Marcel Dekker, 1995. viii + 359 pp. $135.00. Reviewed by Gary Zerbe University of Colorado Health Sciences Center Survival Analysis: A Practical Approach Mahesh K. B. Parmar and Machin David. New York: Wiley, 1995. × + 255 pp. $45. Reviewed by Philip J. Smith Centers for Disease Control and Prevention F. Y. Edgeworth: Writings in Probability, Statistics and Economics, Vols. I–III Charles R. McCann, Jr. (Ed.). Cheltenham, U.K.: Edward Elgar, 1996. xxvi + 454 pp. (Vol. I), viii + 550 pp. (Vol. II), × + 609 pp. (Vol. III). $499.95 (3-volume set). Reviewed by Paul J. Smith University of Maryland at College Park Probability (2nd ed.) A. N. Shiryaev. New York: Springer, 1995. xvi + 621 pp. $62.50. Reviewed by R. Syski University of Maryland A Course in Probability and Statistics Charles J. Stone. Belmont, CA: Duxbury Press, 1996. vi + 838 pp. $85.95. Reviewed by Paul Embrechts ETH-Zurich, Switzerland Subjective Probability George Wright and Peter Ayton (Eds.). Chichester, U.K.: Wiley, 1994. 574 pp. $71.95. Reviewed by David V. Budescu University of Illinois at Urbana-Champaign

A Course in Probability and Statistics.

1. RANDOM VARIABLES AND THEIR DISTRIBUTION Introduction / Sample Distributions / Distributions / Random Variables / Probability Functions and Density Functions / Distribution Functions and Quantiles / Univariate Transformations / Independence 2. EXPECTATION Introduction / Properties of Expectation / Variance / Weak Law of Large Numbers Simulation and the Monte Carlo Method 3. SPECIAL CONTINUOUS MODELS Gamma and Beta Distributions / The Normal Distribution / Normal Approximation and the Central Limit Theorem 4. SPECIAL DISCRETE MODELS Combinatorics / The Binomial Distribution / The Multinomial Distribution / The Poisson Distribution / The Poisson Process 5. DEPENDENCE Covariance, Linear Prediction, and Correlation / Multivariate Expectation / Covariance and Variance - Covariance Matrices / Multiple Linear Prediction / Multivariate Density Function / Invertible Transformations / The Multivariate Normal Distribution 6. CONDITIONAL DISTRIBUTIONS Sampling Without Replacement / Hypergeometric Distribution / Conditional Density Functions / Conditional Expectation / Prediction / Conditioning and the Multivariate Normal Distribution / Random Parameters 7. NORMAL MODELS Introduction / Chi-Square, t, and F Distribution / Confidence Intervals / The t Test of an Inequality / The t Test of an Equality 8. THE F TEST Introduction to Linear Regression / The Method of Least Squares / Factorial Experiments / Input-Response and Experimental Models 9. LINEAR ANALYSIS Linear Spaces / Identifiability / Saturated Spaces / Inner Products / Orthogonal Projections / Normal Equations 10. LINEAR REGRESSION Least-Square Estimation / Sums of Squares / Distribution Theory / sugar Beet Experiment / Lube Oil Experiment / The t Test / Submodels / The F Test 11. ORTHOGONAL ARRAYS Main Effects / Interactions / Experiments with Factors Having Three Levels / Randomization, Blocking, and Covariates 12. BINOMIAL AND POISSON MODELS Nominal Confidence Intervals and Tests / Exact P-values / One-Parameter Exponential Families 13. LOGISTIC REGRESSION AND POISSON REGRESSION Input-Response and Experimental Models / Maximum-Likelihood Estimation / Existence and Uniqueness of the Maximum-Likelihood Estimate / Interactively Reweighted Least-Squares Method / Normal Approximation / The Likelihood-Ratio Test / APPENDICES: A. PROPERTIES OF VECTORS AND MATRICES / B. SUMMARY OF PROBABILITY / C. SUMMARY OF STATISTICS / D. HINTS AND ANSWERS / E. TABLES / INDEX

Crystal Structure, Solid‐state Polymerization, and Ionic Conductivity of Alkali Salts of Unsaturated Carboxylic Acids, 4. Investigations on Lithium Sorbate

AbstractThe structure of lithium sorbate (C6H7LiO2) was determined by single‐crystal X‐ray diffraction. It may be divided into organic and inorganic layers. The ionic part of the structure consists of a two‐dimensional network of corner‐ and edge‐sharing lithium oxotetrahedra, a structural pattern already known from other lithium carboxylates. Irradiation of the substance with X‐rays or its thermal treatment results in the formation of a polymer exhibiting ionic conductivity at higher temperatures. Due to the rather large distances between potentially reactive atoms the polymerization results in a structural breakdown. Nevertheless, during the solid‐state polymerization preferred orientations of building units are partially preserved. The probable course of the polymerization and the structure of the resulting polymer are discussed.