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# Volume 42 - Issue 3 - December 2017

**• Probability models and compounding
** James E. Marengo and David L. Farnsworth

pp. 117–122

**Abstract**

We present the case that the ideas contained in a particular sequence of formulas are important in probability and statistics. The synthesis offered by the concepts in the sequence can be very valuable. Facility with this sequence and its underpinnings should be in the skill set of anyone who uses or studies probability or statistics. For illustrative purposes, we give applications to mixture distributions and Bayesian analyses.

**• The regression line simplified
** Anwar H. Joarder, Munir Mahmood and M. Hafidz Omar

pp. 123–126

**Abstract**

In an elementary statistics course, the concept of the least squares regression line is generally introduced via coordinate geometry and basic algebra. This article renders a simple idea to plot the line of least squares by joining two points, namely the center of gravity of the data, commonly known as the sample mean, and another point based on the mean point. It can be introduced comfortably to high school as well as undergraduate students.

**• Solids of revolution and the Herschel–Maxwell theorem**

R. J. Swift

pp. 127–130

**Abstract **

A characterization of a continuous nonnegative function *f* that is radially symmetric is obtained. The motivation of the problem arises from an elementary consideration of solids of revolution, with the Herschel–Maxwell theorem for the multivariate normal distribution the consequence.

**• A Halley revival: another look at two of his classical gunnery rules
** Robert Kantrowitz and Michael M. Neumann

pp. 131–142

**Abstract**

This article is devoted to a revival of two notable optimization problems from the realm of ballistics that were considered by British astronomer and mathematician Edmond Halley (1656–1742). The first is to determine the angles of launch that maximize the horizontal range of a projectile that lands on a slanted surface, whereas the second is to find angles that minimize the projectile's kinetic energy at launch while ensuring a strike of an intended target. For each of the two problems, we revisit the 'flat Earth model' in which gravity is the only operative force and then enlarge the framework by accounting for drag that is linear or quadratic in speed. The unifying thread is an unassuming trigonometric function that sheds light on the duality to Halley's problems and helps reveal analytic and geometric parallels between the optimal flight curves in the nonresistive environment and that in which air resistance is quadratic in speed.

**• A graph-theoretic approach to Jacobsthal polynomials
** Thomas Koshy and Martin Griffiths

pp. 143–150

**Abstract**

Using a graph-theoretic approach, we obtain several identities involving both Jacobsthal and Jacobsthal–Lucas polynomials. The main focus is with respect to the method of proof rather than with the results themselves. Although the results presented are not new, they are derived here via the enumeration of walks in digraphs. We employ two different digraphs to accomplish this, one with two vertices and the other with three.

**• A digraph model for the Jacobsthal family****
** Thomas Koshy

pp. 151–156

**Abstract**

We construct a graph-theoretic model for Jacobsthal and Jacobsthal–Lucas numbers. As byproducts, we then develop addition formulas for the Jacobsthal family.

**• ****A moment closure technique for a stochastic predator–prey model ****
** Diana Curtis and Jennifer Switkes

pp. 157–168

**Abstract**

The classical deterministic Lotka–Volterra predator–prey model famously leads to closed curves in the predator–prey phase plane. A stochastic version of this model has the form of a simple birth–death process, with the expected values governed by a system of differential equations almost identical in form to the deterministic system, the difference in rate function for each species being proportional to the time-dependent covariance of the populations of the two species. We explore the impact of this covariance term. Assuming that the distribution of the two populations is roughly multivariate normal, we use a moment closure technique to obtain a closed system of differential equations for the expected values, variances, and covariance of the populations.

**• Letter to the Editor: A remark on the immigration–catastrophe process
** J. Gani and R. J. Swift

pp. 169–170

**• Index to Volume 42
** p. 171