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Mandelbrot and Fat Tails
The Mandelbrot set is the set of complex numbers c such that iterations of f(z) = z² + c remain bounded. But how do you know an iteration will remain bounded? You know when it becomes unbounded—if |z| > 2 then the point isn't coming back—but how do you know whether an iteration will never become unbounded? You
Vanity Bitcoin addresses
Bitcoin addresses are essentially hash values of public keys encoded in Base58. More details here. The addresses are essentially random characters chosen from the Base58 alphabet: uppercase and lowercase Latin letters and digits, with 0 (zero), I (capital I), O (capital O), and l (lowercase l) removed to prevent errors. You could create an address
An integral theorem of Gauss
Gauss proved in 1818 that the value of integral is unchanged if x and y are replaced by (x + y)/2 and √(xy), i.e. if you replaced x and y with their arithmetic mean and geometric mean [1]. So, for example, if you wanted to compute you could instead compute Notice that the coefficients of sin² θ and cos² θ
You can’t have everything you want: beta edition
The beta distribution is a conjugate prior for a binomial likelihood function, so it makes posterior probability calculations trivial: you simply add your data to the distribution parameters. If you start with a beta(α, β) prior distribution on a proportion θ, then observe s successes and f failures, the posterior distribution on θ is beta(α +