What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 12

Let X_1, X_2, X_3, \dots be a sequence of independent and identically distributed random variables, and let H_n be the number of “record highs” upon to and including event n. For example, each X_i can represent the amount of rainfall in a year, where X_1 is amount of rainfall recorded the first time that records were kept. As shown in Gamma (page 125), the expected number of record highs is

H_n = \displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}.

As noted in Gamma,

Two arbitrary chosen examples are revealing. The Radcliffe Meteorological Station in Oxford has data for rainfall in Oxford between 1767 and 2000 and there are five record years; this is a span of 234 recorded years and H_{234} = 6.03. For Central Park, New York City, between 1835 and 1994 there are six record years over the 160-year period and H_{160} = 5.65, providing good evidence that English weather is that bit more unpredictable.

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

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