David L. Farnsworth

Testing for Mutual Exclusivity

A test for two events being mutually exclusive is presented for the case in which there are known rates of misclassification of the events. The test can be utilized in other situations, such as to test whether a set is a subset of another set. In the test, the null value of the probability of the intersection is replaced by the expected value of the number determined to be in the intersection by the imperfect diagnostic tools. The test statistic is the number in a sample that is judged to be in the intersection. Medical testing applications are emphasized.

Transformation of power means and a new class of means

Abstract Using transformations associated with power means, a new class of means is defined. The behavior of these means is examined vis-a-vis power means, as a free parameter is varied. In particular, limits which contain those of J. L. Brenner (Pi Mu Epsilon J. 8(1985), 160–163) and L. Hoehn and I. Niven (Math. Mag. 58 (1985), 151–156) are obtained.

A Geometric Derivation of the Irwin-Hall Distribution

The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

The Effect of a Single Point on Correlation and Slope

By augmenting a bivariate data set with one point, the correlation coefficient and/or the slope of the regression line can be changed to any prescribed values. For the target value of the correlation coefficient or the slope, the coordinates of the new point are found as a function of certain statistics of the original data. The location of this new point with respect to the original

An introduction to Minkowski geometries

The fundamental ideas of Minkowski geometries are presented. Learning about Minkowski geometries can sharpen our students’ understanding of concepts such as distance measurement. Many of its ideas are important and accessible to undergraduate students. Following a brief overview, distance and orthogonality in Minkowski geometries are thoroughly discussed and many illustrative examples and applications are supplied. Suggestions for further study of these geometries are given. Indeed, Minkowski geometries are an excellent source of topics for undergraduate research and independent study.

The Behavior of Midsets When Repeatedly Taking the Midset of Two Lines in Geometry

We study the outcome of taking midsets of two lines in geometry. We establish the algorithm for repeatedly finding these midsets and characterize the limiting midsets. We discuss the issue of angle measurement in Minkowski geometries, especially with respect to the limiting midsets.

Student presentations in the classroom

For many years, the author has been involving his students in classroom teaching of their own classes. The day-to-day practice is described, and the advantages and disadvantages for both the instructor and the students are discussed. Comparisons with the Moore Method of teaching are made.

A Calculus Theorem Motivated by a Statistics Problem

two digits. For example, 1/97 = .01030927 ..., the powers of three. The Fibonacci numbers result from p = q = 1, co = 0, cl = 1, x = 10 in (2) as 10/89 = .11235 ... or using x = 100, we find 100/9899 = .0101020305081321 .... If we make up our own sequence 2, 5, 7, 12, 19,..., where p = q = 1, co = 3, cl = 2, and let x = 100, we find 203/9899 = .0205071219.... We can put an arithmetic progression into the digits of a decimal expansion by taking cl = a, C2 a + d, co = a d. Then Cn+2 = 2cn+1 cn, and p = 2, q = -1 in (2). Let x = 102, and expand fractions y/9801. The terms of the chosen arithmetic progression appear in blocks of two digits. Some examples are:

A Cautionary Note Concerning the Cox and Stuart Test

A homework assignment led to the observation that the Cox and Stuart test is not symmetric under the transposition of the two variables. Examples of this feature are presented.

The Case Against Histograms

A homework assignment used to show students that histograms can be misleading is presented.