  In this paper we consider functions of the type $$f(x) = \sum_{n=0}^\infty
a_n g(b_nx+\theta_n),$$ where $(a_n)$ are independent random variables
uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \geq b
>1$, $a^2b> 1$ and $g$ is a $C^1$ periodic real function with finite number of
critical points in every bounded interval. We prove that the occupation measure
for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension
of the graph of $f$ is almost surely equal to $D = 2+ \log{a}/\log{b}$ provided
$ b = \lim_{n\rightarrow \infty}b_{n+1}/b_n>1$ and $ab>1$.
