Let $f(z)=a_0+a_1z+a_2z^2+\cdots$ be an analytic function over the unit disk in the complex plane. Let $$Rf(z)=X_0 a_0 + X_1 a_1 z + X_2 a_2 z^2+ \cdots$$ be its randomization, where $X_0, X_1, X_2, \cdots$ is an i.i.d. sequence of symmetric random variables. We show that there exists a well-defined notion of polynomial growth rate for random analytic functions $Rf$. We characterize those $f(z)$, in terms of coefficients, such that $Rf$ has a polynomial growth rate almost surely. Then we show that the rate of $Rf$ is improved when compared with that of $f$, and the order of improvement is at most ½. (Joint work with Pham Trong Tien at Vietnam National University, Hanoi)