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# Volume 37 - Issue 2 - December 2012

### The Cauchy-Frullani integral formula extended to double integrals

Brita Jung

Abstract

In this note we study the development of the integral formula commonly known as the Cauchy-Frullani integral. It is then shown how a similar formula can be obtained for a double integral, subject to some conditions.

### How Euler found the eccentricity of planetary orbits

Thomas J. Osler and Jasen Andrew Scaramazza

Abstract

Euler discovered an interesting method for astronomers to calculate the eccentricity of a celestial body in an elliptical orbit. We describe the mathematics behind Euler's method and show how it can be used by astronomical observers.

### On the statistical independence of primes

Kais Hamza and Fima Klebaner

Abstract

We characterise probability measures on the set of integers under which the events of being divisible by different prime numbers are independent. Classical identities, such as the Euler product representation, can then easily be obtained by probabilistic means.

### Binomial prediction using the frequent outcome approach

B. O'Neill

Abstract

Within the context of the binomial model, we analyse sequences of values that are almost-uniform and we discuss a prediction method called the frequent outcome approach, in which the outcome that has occurred the most in the observed trials is the most likely to occur again. Using this prediction method we derive probability statements for the prior probability of correct prediction, conditional on the underlying parameter value in the binomial model. We show that this prediction method converges to a level of accuracy that is equivalent to ideal prediction based on knowledge of the model parameter.

### The duplication formula derived from the normal distribution

Rasul A. Khan

Abstract

The Legendre duplication formula for the gamma function is derived from the normal distribution. Its connection with the binomial distribution is also discussed. A classical integral formula in terms of gamma functions is obtained as a byproduct of the normal derivation.

### The smallest upper bound for the *p*th absolute central moment of a class of random variables

M. Egozcue, L. F. García, W-K. Wong and R. Zitikis

Abstract

We establish the smallest upper bound for the *p*th absolute central moment over the class of all random variables with values in a compact interval. Numerical values of the bound are calculated for the first ten integer values of *p*, and its asymptotic behaviour derived when *p* tends to infinity. In addition, we establish an analogous bound in the case of all symmetric random variables with values in a compact interval. Such results play a role in a number of areas including actuarial science, economics, finance, operations research, and reliability.

### Kelly gambling with the stock market and banks

Ravi Phatarfod

Abstract

In this paper we consider the relative merits of putting money in a bank with a fixed compound interest as against investing it in an investment fund with exposure to the stock market. We show that if the fund manager adopts the Kelly gambling criterion, then there is an upper bound to the volatility of this fund beyond which investing in it is not as profitable as putting money in a bank. We assume a variety of distributions for the random change in the yearly capital value of the investment.

### On individual neutrality and collective decision making

Mu Zhu, Shangsi Wang and Lu Xin

We derive a simple mathematical `theory' to show that two decision-making entities can work better together *only if* at least one of them is occasionally willing to stay neutral. This provides a mathematical `justification' for an age-old cliché among marriage counselors.