The doctrine of chances or A method of calculating the probability of events in play

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On the length of the longest head run

We evaluate the accuracy of approximation to the distribution of the length of the longest head run in a Markov chain with a discrete state space. An estimate of the accuracy of approximation in terms of the total variation distance is established for the first time

Why do lifespan variability trends for the young and old diverge? A perturbation analysis

Background: Variation in lifespan has followed strikingly different trends for the young and old: while overall lifespan variability has decreased as life expectancy at birth has risen, the variability conditional on survival to older ages has increased. These diverging trends reflect changes in the underlying demographic parameters determining age-specific mortality. Objective: We ask why the variation in the adult ages at death has followed a different trend than the variation at younger ages, and aim to explain the diverging patterns in terms of historical changes in the age schedule of mortality. Methods: Using simulations, we show that the empirical trends in lifespan variation are well characterized using the Siler model, which describes the mortality hazard across the full lifespan using functions representing early-life, later-life, and background mortality. We then obtain maximum likelihood estimates of the Siler parameters over time. Finally, we express lifespan variation in terms of a Markov chain model, and apply matrix calculus perturbation analysis to compute the sensitivity of age-specific lifespan variance trends to the changing Siler model parameters. Results: Our analysis produces a detailed quantification of the impact of changing demographic parameters on the pattern of lifespan variability at all ages, highlighting the impact of declining childhood mortality on the reduction of lifespan variability and the impact of improved survival in adulthood on the rising variability of lifespans at older ages. Conclusions: These findings provide insight into the dynamic relationship between the age pattern of survival improvements and time trends in lifespan variability

De Moivre's Poisson Approximation to the Binomial

In his first work on probability, written in 1711, Abraham De Moivre looked at the problem of finding the number of trials required in a binomial experiment to achieve a probability of 1/2 of finding at least some given number of successes. He looked at two cases: when the probability of success "p" = 1/2 and when "p" is small but "n", the number of trials, is large. In the latter case, unlike other problems that he solved in probability, De Moivre never revealed his method of solution. We explore the solution that De Moivre originally suggests and find that his method does not work. We explore other numerical solutions and put forward the suggestion that De Moivre relied on a very cumbersome and tedious method of solution based on his earlier work on series in the 1690s. Since his method was neither quick nor mathematically elegant, he never revealed the method that he used to obtain his numerical solutions. Copyright (c) 2009 The Authors. Journal compilation (c) 2009 International Statistical Institute.

The Doctrine of Chances : Or, A Method of Calculating the Probabilities of Events in Play

By A. De Moivre...Supralibros der Naturforschenden Gesellschaft in Zürich Exemplar der ZB Züric