For more information on how you can subscribe to our journals please read the information on our subscriptions page.

Click here for information on submitting papers to the Applied Probability Trust.

# Volume 35 - Issue 1 - June 2010

**• Karl Pearson a century and a half after his birth**Oscar Sheynin

**pp. 1–9**

**Abstract **

In this biography of Karl Pearson I describe his transition from history to mathematics, statistics, and eugenics and his work as the leader of the Biometric School and creator of biometry. I also point out his interesting studies of physics and describe how other scientists regarded him. I especially note that Soviet statisticians did not recognize Pearson's merits.

**• An explicit solution to the problem of optimizing the allocations of a **

** bettor's wealth when wagering on horse races
** Peter Smoczynski, Dave Tomkins

pp. 10–17

**Abstract**

An explicit formula for the optimal strategy for betting allocation on horse races is given. The formula for the maximal value of the logarithm of average geometric growth rate is also given. The solution is obtained with the help of KKT theory. Application of the formulas requires the optimal set of horses to bet on to be constructed. For a horse to be included in the set, the expected revenue rate must be greater than the fraction of unallocated wealth. A simple way, without solving any equations, for determining the optimal set of horses is given.

**• Rediscoveries of a first passage time result for dams and shot noise****
** Geoffrey F. Yeo

pp. 18–22

**Abstract**

This note explores the history of a first passage time result in dam theory and shot noise processes. This result has appeared in various forms both in the same and in different journals and has been 'rediscovered' several times, without some authors being aware of its prior existence.

**• A simple Parrondo paradox****
** Michael Stutzer

pp. 23–28

**Abstract**

The Parrondo paradox is a counterintuitive probabilistic phenomenon in situations of repeated decision making with uncertainty. It can be simply stated as follows: given two games, each with a higher probability of losing than winning, it is possible to construct a winning strategy by playing the games alternately. The best-known and analyzed examples are somewhat involved, and may not be relevant to practical situations of repeated decision making, e.g. casino games and investment in securities markets. Here, we construct a very simply stated and analyzed example of the Parrondo paradox, using a model of stylized Blackjack-type betting that is well-known in the gambling literature.

**• On the Kermack-McKendrick approximation for removals from an **

** epidemic
** J. Gani, R. J. Swift

pp. 29–36

**Abstract**

The Kermack-McKendrick model for the SIR epidemic is re-examined with particular attention to the exponential approximation used by them to estimate the final number of removals. It is shown that while this approximation is valid when the initial number

*x*

_{0}of susceptibles is close to

*p*, the ratio of the removal to the infection parameters, it fails when

*x*

_{0}>

*p*. In all cases the exact number of removals is greater than the approximate number.

**• A Black-Scholes model with GARCH volatility****
** H. Gong, A. Thavaneswaran, J. Singh

pp. 37–42

**Abstract**

Option pricing based on the Black-Scholes model is typically obtained under the assumption that the volatility of the return is a constant. In this paper, we develop a new method for pricing derivatives under the Black-Scholes model with GARCH volatility by viewing the call price as an expected value of a truncated normal distribution. Using return data, we estimate the mean, variance, and kurtosis of the random volatility in the Black-Scholes model. An extensive empirical analysis of the European call option valuation of the S&P 100 Index shows that our method outperforms other competing GARCH pricing models and the pricing errors when using our method are quite small even though our estimation procedure is based only on the historical return data.

**• The geometry of undamped harmonic oscillators****
** Yousef Daneshbod, Joe Latulippe

pp. 43–53

**Abstract**

In this article, we present a geometrical approach to understanding undamped harmonic oscillators. We investigate the phase-plane behavior of both unforced and forced oscillations. The methods explained in this article are not commonly found in differential equation texts and provide an illustrative way for understanding harmonic oscillators. A conservation of energy approach and a substitution method that allows solution patterns to be examined without explicitly having to find the time series is presented.

**• Letter to the Editor: A combinatorial proof of a novel binomial identity****
** Gérard C. Nihous

pp. 54–56

**• Letter to the Editor: On Borromean rings****
** Helge Tverberg

pp. 57–60

**• Letter to the Editor: A generalization of the Borel****–Cantelli lemma****
** Narayanaswamy Balakrishnan, Alexei Stepanov

pp. 61–62