  We consider the skew-product of an expanding map $E$ on the circle $\mathbb
T$ with an almost surely $\mathcal C^k$ random perturbation
$\tau=\tau_0+\delta\tau$ of a deterministic function $\tau_0$: \[F
:\left\{\begin{array}{rcl} \mathbb T \times \mathbb R & \longrightarrow &
\mathbb T \times \mathbb R\\ (x,y)& \longmapsto & (E(x), y+\tau(x))\\
\end{array} \right.\] The associated transfer operator $\mathcal L:u \in
\mathcal C^k (\mathbb T \times \mathbb R) \mapsto u\circ F$ can be decomposed
with respect to frequency in the $y$ variable into a family of operators acting
on functions on the circle: \[\mathcal L_\xi :\left\{\begin{array}{rcl}
\mathcal C^k(\mathbb T) & \longrightarrow & \mathcal C^k(\mathbb T)\\ u &
\longmapsto & e^{i\xi\tau}u\circ E \\ \end{array} \right.\] We show that the
flat traces of $\mathcal L^n_{\xi}$ behave as normal distributions in the
semiclassical limit $n, \xi\to\infty$ up to the Ehrenfest time $n\leq
c_k\log\xi$.
