The Koebe function is defined on the unit circle of the complex plane by: \[f(x) = \frac{x}{(1 - x)^2}.\] Its Maclaurin series expansion tells that \[f(x) = \sum_{n = 1}^{\infty} n x^n,\] which allows us to interpret the Koebe function as the expected value of the following random variable \(Y\).
Assume that \(X_i\) is an infinite Bernoulli process with the probability that \(X_i = 1\) is \(p.\) Regarding its realization as a sequence of 0s and 1s, group the input in non-overwrapping consective bits of length \(1, 2, 3,\) etc. in this order. For example, if the input sequence is 011010111101100..., then grouping looks like (0)(11)(010)(1111)(01100).... Then discard the groups containing 0; e.g. the final output of the above example is (11)(1111).... Call the length of the remaining 1s \(Y\). Then \[\mathbb{E}[Y] = f(p).\]