  We determine in this paper the distribution of the number of points on the
cyclic covers of $\mathbb{P}^1(\mathbb{F}_q)$ with affine models $C: Y^r =
F(X)$, where $F(X) \in \mathbb{F}_q[X]$ and $r^{th}$-power free when $q$ is
fixed and the genus, $g$, tends to infinity. This generalize the work of
Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered
different families of curves over $\mathbb{F}_q$. In all cases, the
distribution is given by a sum of random variables.
