John D. Cook

Suppose you stand at 0 and flip a fair coin. If the coin comes up heads, you take a step to the right. Otherwise you take a step to the left. How much of the time will you spend to the right of where you started? As the number of steps N goes to infinity, […]

Take a real number x and expand it as a continued fraction. Compute the geometric mean of the first n coefficients. Aleksandr Khinchin proved that for almost all real numbers x, as n → ∞ the geometric means converge. Not only that, they converge to the same constant, known as Khinchin’s constant, 2.685452001…. (“Almost all” […]

Greg Egan’s short story Glory features a “xenomathematician” who discovers that an ancient civilization had produced a sort of grand unification of their various branches of mathematics. It was not a matter of everything in mathematics collapsing in on itself, with one branch turning out to have been merely a recapitulation of another under a different […]

Akin’s eighth law of spacecraft design says In nature, the optimum is almost always in the middle somewhere. Distrust assertions that the optimum is at an extreme point. When I first read this I immediately thought of several examples where theory said that an optima was at an extreme, but experience said otherwise. Linear programming (LP) says […]

Dose-finding trials of chemotherapy agents look for the MTD: maximum tolerated dose. The idea is to give patients as much chemotherapy as they can tolerate, hoping to do maximum damage to tumors without doing too much damage to patients. But “maximum tolerated dose” implies a degree of personalization that rarely exists in clinical trials. Phase I […]

Sometimes it’s rational to walk away from something you’ve invested a great deal in. It’s hard imagine how investors could abandon something as large and expensive as a shopping mall. And yet it must have been a sensible decision. If anyone disagreed, they could buy the abandoned mall on the belief that they could make a […]

This week’s resource post lists some articles along with source code I’ve posted on CodeProject. Probability Pitfalls in Random Number Generation includes several lessons learned the hard way. Simple Random Number Generation is a random number generator written in C# based on George Marsaglia’s WMC algorithm. Finding probability distribution parameters from percentiles Numerical computing Avoiding […]

I’ve started a new Twitter account @PerlRegex for Perl regular expressions. My original account, @RegexTip, is for regular expressions in general and doesn’t go into much detail regarding any particular implementation. @PerlRegex goes into the specifics of regular expressions in Perl. Why specifically Perl regular expressions? Because Perl has the most powerful support for regular […]

In mathematics, log means natural logarithm by default; the burden of explanation is on anyone taking logarithms to a different base. I elaborate on this a little here. Looking through Andrew Gelman and Jennifer Hill’s regression book, I noticed a justification for natural logarithms I hadn’t thought about before. We prefer natural logs (that is, logarithms […]

Every Wednesday I’ve been pointing out various resources on my web site. So far they’ve all been web pages, but the following are all PDF files. Probability and statistics: How to test a random number generator Predictive probabilities for normal outcomes One-arm binary predictive probability Relating two definitions of expectation Illustrating the error in the […]

A bigram is a pair of letters. For various reasons—word games, cryptography, user interface development, etc.—people are interested in knowing which bigrams occur most often, and so such information is easy to find. But sometimes you might want to know which bigrams occur least often, and that’s harder to find. My interest is finding safe […]

Suppose you have data in an N-dimensional space where N is large and consider the cube [-1, 1]N. The coordinate basis vectors start in the center of the cube and poke out through the middle of the faces. The diagonals of the cube run from the center to one of the corners. If your points cluster along one […]