The uniformly most accurate (UMA) is an important optimal approach in interval estimation, but the current literature often introduces it in a confusing way, rendering the learning, teaching and researching of UMA problematic. Two major aspects cause this confusion. First, UMA is often interpreted to maximize the accuracy of coverage, but in fact,…

Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability…

It is well known that two events A and B are independent if and only if . Here, we derive a condition for independence not well known. We then extend the condition for independence of n events.

We consider here two questions posed by first-year undergraduate students in a statistics tutorial session. These questions are related, and each concerns an aspect of independence in probability. The notion of independence does tend to be a little less intuitive than some of the other elementary probabilistic concepts, thereby providing students…

The treatment of kurtosis in textbooks is both sparse and contradictory with applications rarely discussed. To address this, an easily understood definition of kurtosis is introduced and important applications are demonstrated. Two different approaches to teaching kurtosis are presented based on a financial application.

A number of probabilistic experiments are described to estimate e, p and v2, with results from computer simulations being used to investigate convergence. A number of possible classroom exercises and extensions are presented.

The mathematics education literature shows that encouraging students to develop multiple solutions for given problems has a positive effect on students' understanding and creativity. In this paper, we present an example of multiple-solution problems in statistics involving a set of non-traditional dice. In particular, we consider the exact…

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using "induction", "polynomial identities" nor "generating functions", and will give a proof of the "Vandermonde…

In this paper, we derive the result of the classical gambler's ruin problem using elementary linear algebra. Moreover, the pedagogical advantage of the derivation is briefly discussed.

We consider here the problem of calculating the moments of binomial random variables. It is shown how formulae for both the raw and the central moments of such random variables may be obtained in a recursive manner utilizing Stirling numbers of the first kind. Suggestions are also provided as to how students might be encouraged to explore this…

This article shows a generalization of Galileo's "passe-dix" game. The game was born following one of Galileo's [G. Galileo, "Sopra le Scoperte dei Dadi" (Galileo, Opere, Firenze, Barbera, Vol. 8). Translated by E.H. Thorne, 1898, pp. 591-594] explanations on a paradox that occurred in the experiment of tossing three fair "six-sided" dice.…

Since 1986, Marilyn Vos Savant, who is listed in the "Guinness Book of World Records Hall of Fame" for the highest IQ, has had a weekly column that is published in "Parade Magazine." In this column, she answers readers' questions on a wide variety of subjects including mathematics and particularly probability. Many of the mathematically oriented…

This article describes some misconceptions about random variables and related counter-examples, and makes suggestions about teaching initial topics on random variables in general form instead of doing it separately for discrete and continuous cases. The focus is on post-calculus probability courses. (Contains 2 figures.)

This article deals with a brief biographical sketch of Joseph Fourier, his first celebrated work on analytical theory of heat, his first great discovery of Fourier series and Fourier transforms. Included is a historical development of Fourier series and Fourier transforms with their properties, importance and applications. Special emphasis is made…

This classroom note illustrates how dynamic visualization can be used to teach conditional probability and Bayes' theorem. There are two features of the visualization that make it an ideal pedagogical tool in probability instruction. The first feature is the use of area-proportional Venn diagrams that, along with showing qualitative relationships,…