On distributions computable by random walks on graphs

We answer a question raised by Donald E. Knuth and Andrew C. Yao, concerning the class of polynomials on [0,1] that can be realized as the distribution function of a random variable, whose binary expansion is the output of a finite state automaton driven by unbiased coin tosses. The polynomial distribution functions which can be obtained in this way are precisely those with rational coefficients, whose derivative has no irrational roots on [0,1]. We also show, strengthening a result of Knuth and Yao, that all smooth distribution functions which can be obtained by such automata are polynomials.