Minkowski's question-mark function
In mathematics, Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904.[1] It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938.[2] It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.
Probability distribution[edit]
Restricting the Minkowski question mark function to ?:[0,1] → [0,1], it can be used as the cumulative distribution function of a singular distribution on the unit interval. This distribution is symmetric about its midpoint, with raw moments of about m1 = 0.5, m2 = 0.290926, m3 = 0.186389 and m4 = 0.126992,[13] and so a mean and median of 0.5, a standard deviation of about 0.2023, a skewness of 0, and an excess kurtosis about −1.147.