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# Volume 34 - Issue 1 - June 2009

**• Obituary: Kiyoshi Itô****
** Randall Swift

pp. 1–3

**• Extending Halley's problem: firing a mortar when there is air**

**resistance**

C. W. Groetsch

pp. 4–10

**Abstract**

Halley’s classical results on minimum energy trajectories are extended to linearly resisting media. The analysis relies on a simple fixed point theorem and implicit relationships, and leads to an effective numerical method for computing the optimal launch angle.

**• The contribution of Sir Ronald Ross to epidemic modelling****
** J. Gani

pp. 11–14

**Abstract**

A brief sketch of the life and work of Sir Ronald Ross is provided, and his papers on

*a priori pathometry*summarized. Ross was one of the earliest epidemic modellers.

**• Repeated subdivision of triangles by their medians****
** Nelson M. Blachman

pp. 15–19

**Abstract**

It is shown that repeated subdivision of triangles by their medians into six smaller triangles tends to yield increasingly thin triangles having lognormally distributed thicknesses whose geometric-mean value approaches zero as the subdivision proceeds, a triangle's `thickness' being defined as the ratio of its shortest altitude to its longest side.

**• An alternative derivation of the multi-parameter Cramér–Rao **

** inequality
** Brandi A. Greer, James D. Stamey, Dean M. Young, David J. Ryden

pp. 20–24

**Abstract**

We derive the multi-parameter Cramér–Rao inequality using only multivariate calculus, and an elementary property of partitioned positive-definite matrices. The derivation is straightforward and brief, and assumes only the commutativity of a vector derivative and a multiple integral. Thus, the new proof has several advantages over the proofs given by Shalaevskii (1961), Linnik (1961), Rao (1973), Fabian and Hannan (1977), Mardia, Kent and Bibby (1979), Witting (1985), Drygas (1987), and Kagan (2001).

**• Limits of mean and variance of a Fibonacci distribution****
** David K. Neal

pp. 25–29

**Abstract**

A probability distribution is defined on the set {1,...,

*n*} with decreasing masses determined by the Fibonacci sequence

*F*,...,

_{n}*F*. The mean and variance are computed and it is shown that the limit of these parameters, as $n$ tends to infinity, are simple functions of the golden ratio, \Phi.

_{1}**• A strategy that exploits (unknown) bias in tossing a real coin****
** Stephen J. Royle

pp. 30–33

**Abstract**

Interest in the game of tossing a coin dates back many centuries. The French mathematician Pierre–Simon Laplace (1749–1827) was one who notably made early contributions using probability theory and its wider application. In this paper we consider a strategy that provides a positive expectation when wagering on either heads or tails on the toss of a coin. It is shown how the prior event of the coin landing on heads or tails can be useful in enhancing the player's chances in the game. The coin is real, and so we make no assumption as to its fairness. Indeed, it is probably biased due to its manufacture, weight, wear, and other mechanical factors. Moreover, bias can be introduced into the mix by the human dealer, who may have a characteristic coin toss action.

**• Negative moments for the conditional binomial****
** Christopher S. Withers, Saralees Nadarajah

pp. 34–42

**Abstract**

Recently, several authors have been interested in obtaining asymptotic expansions for the negative moments of the binomial conditioned to be positive. Here, we show how asymptotic expansions for its moments and cumulants can be easily derived. As an aside, we also give exact results for the mean of the inverse of a binomial plus an integer.

**• ****Constrained confidence estimation of the binomial p via tail functions**

**Borek Puza, Terence O'Neill**

pp. 43–48

**Abstract**

A methodology is proposed for `exact' confidence estimation of the binomial parameter

*p*when that parameter is constrained. It is shown how the technique of tail functions can be used to construct a suitable generalised Clopper–Pearson confidence interval when

*p*is known to lie between two bounds, and how the interval can be engineered for optimality in terms of prior expected length. An example is provided which illustrates the applicability of the theory to gambling.

**• ****Letter to the Editor: Birthday problem that includes a leap day****
** Mohammed R. Karim

pp. 49–50

**•**

**Letter to the Editor: An interesting property of finite birth and death**

**processes**

P. R. Parthasarathy

pp. 51–53

**•**

**Letter to the Editor: An identity for the bivariate normal**

**Christopher S. Withers, Saralees Nadarajah**

pp. 54–56

**•**

**Letter to the Editor: Constants in a differentiation process that become**

**variables**

Haralds Petersons

pp. 57–58