A class of singular continuous functions (Q2367755)

The author constructs singular functions as follows. Let \(X_ 1\) be a random variable taking the values \(0,1,\dots,N- 1\) according to the probability vector \({\mathbf p}\). If \((X_ n)_{n\in\mathbb{N}}\) is a sequence of independent and identically as \(X_ 1\) distributed random variables then the distribution function \(F(X;{\mathbf p})\) of \(X= \sum^ \infty_{n=1} {X_ n\over N^ n}\) is a singular function except \({\mathbf p}\) is uniform.