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 study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…

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…

In Probability and Statistics taught to mathematicians as a first introduction or to a non-mathematical audience, joint independence of events is introduced by requiring that the multiplication rule is satisfied. The following statement is usually tacitly assumed to hold (and, at best, intuitively motivated): If the n events E[subscript 1],…

The linear regression model is one of the most popular models in statistics. It is also one of the simplest models in statistics. It has received applications in almost every area of science, engineering and medicine. In this article, the authors show that adding a predictor to a linear model increases the variance of the estimated regression…

Moments and cumulants are expressed in terms of each other using Bell polynomials. Inbuilt routines for the latter make these expressions amenable to use by algebraic manipulation programs. One of the four formulas given is an explicit version of Kendall's use of Faa di Bruno's chain rule to express cumulants in terms of moments.

Research on teaching high school mathematics shows that the topic of percentages often causes learning difficulties. This article describes a method of teaching percentages that the authors used in university bridging courses. In this method, the information from a word problem about percentages is presented in a two-way table. Such a table gives…

In this note the Nearest Neighbour Index is investigated in three cases, linear, 2-dimensional and 3-dimensional. In each case the formula is investigated and the numerical values for the data points to be viewed as attracting each other, repelling each other or being randomly distributed are justified. Also, in each of the three cases mentioned…

Extreme-value statistics is often used to estimate so-called return values (actually related to quantiles) for environmental quantities like wind speed or wave height. A basic method for estimation is the method of block maxima which consists in partitioning observations in blocks, where maxima from each block could be considered independent.…

The mean and variance of some continuous distributions, in particular the exponentially decreasing probability distribution and the normal distribution, are considered. Since they involve integration by parts, many students do not feel comfortable. In this note, a technique is demonstrated for deriving mean and variance through differential…

In this paper a more accurate index of asymmetry is studied. Specifically, the use of the left and right variance is proposed and an index of asymmetry based on them is introduced. Several examples demonstrate its usefulness. The question of evaluating more accurately the dispersion of data about the average emerges in all non-symmetric…

In insurance, the analyst is often faced with a large number of inter-related variables for which correlations need to be estimated. Clearly, all correlations lie in the interval [-1, 1], but the numbers cannot be assigned independently. Here, the choices left to the analyst are considered from both a geometric and a probabilistic viewpoint. In…

Given a set of urns, each filled with a mix of black chips and white chips, what is the probability of drawing a black chip from the last urn after some sequential random shifts of chips among the urns? The Total Probability Formula (TPF) is the common tool to solve such a problem. However, when the number of urns is more than two and the number…

It is very well known that the Cauchy-Schwarz inequality is an important property shared by all inner product spaces and the inner product induces a norm on the space. A proof of the Cauchy-Schwarz inequality for real inner product spaces exists, which does not employ the homogeneous property of the inner product. However, it is shown that a real…

The traditional approach to expressing cumulants in terms of moments is by expansion of the cumulant generating function which is represented as an embedded power series of the moments. The moments are then obtained in terms of cumulants through successive reverse substitutions. In this note we demonstrate how cumulant-moment relations are…