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Christopher Long
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"Tick, clong, tick, clong, tick, clong, went the night." - Thurber
"Tick, clong, tick, clong, tick, clong, went the night." - Thurber

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Poisson Games and Sudden-Death Overtime
Let's say we have a game that can be reasonably modeled as two independent Poisson processes with team \(i\) having parameter \(\lambda_i\). If one team wins in regulation with team \(i\) scoring \(n_i\), then it's well-known we have the MLE estimate \(\hat...

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Why does Kaggle use Log-loss?
If you're not familiar with Kaggle , it's an organization dedicated to data science competitions to both provide ways to companies to potentially do cheaper analytics, as well as to identify talented data scientists. Competitions are scored using a variety ...

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The Kelly Criterion and a Sure Thing
The Kelly Criterion is an alternative to standard utility theory, which seeks to maximize expected utility. Instead, the Kelly Criterion seeks to maximize expected growth . That is, if we start out with an initial bankroll \(B_0\), we seek to maximize \(\ma...

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Prime Divisors of \(3^{32}-2^{32}\)
Find four prime divisors < 100 for \(3^{32}-2^{32}\).
Source: British Math Olympiad, 2006.

This factors nicely as \(3^{32}-2^{32} = \left(3^{16}+2^{16}\right)\left(3^{16}-2^{16}\right)\), and we can continue factoring in this way to get \[3^{32}-2^{32} = \...

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Highest Powers of 3 and \(\left(1+\sqrt{2}\right)^n\)
Let \(\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}\), where \(a\) and \(b\) are integers. What is the greatest common divisor of \(b\) and \(81\)? Source: 2011-2012 SDML High School 2a, problem 15. Let \((1+\sqrt{2})^n = a_n + b_n \sqrt{2}\). I've thought abo...

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Sum of Two Odd Composite Numbers
What is the largest even integer that cannot be written as the sum of two odd composite numbers? Source: AIME 1984, problem 14. Note \(24 = 3\cdot 3 + 3\cdot 5\), and so if \(2k\) has a representation as the sum of even multiples of 3 and 5, say \(2k = e_3\...

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What's the Value of a Win?
In a previous entry I demonstrated one simple way to estimate an exponent for the Pythagorean win expectation . Another nice consequence of a Pythagorean win expectation formula is that it also makes it simple to estimate the run value of a win in baseball,...

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A Simple Estimate for Pythagorean Exponents
Given the number of runs scored and runs allowed by a baseball team, what's a good estimate for that team's win fraction? Bill James famously came up with what he called the " Pythagorean expectation " \[w = \frac{R^2}{R^2 + A^2},\] which can also be writte...

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Behind the Speadsheet
In the book "The Only Rule Is It Has to Work: Our Wild Experiment Building a New Kind of Baseball Team" , Ben Lindbergh and Sam Miller recount a grand adventure to take command of an independent league baseball team, with the vision of trying every idea, sa...

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When is a Lead Safe in the NBA?
Assuming two NBA teams of equal strength with \(t\) seconds remaining, what is a safe lead at a prescribed confidence level? Bill James has a safe lead formula for NCAA basketball , and the topic has been addressed by other researchers at various levels of ...
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