In this thesis, we will give an asymptotic behavior of the number of hyperelliptic curves with Weierstrass points of arbitrary genus $g$ over $F_q$ when $q$ is even. Our result is
$2q^{2g−1}+q^{g−1}+O(q^{g−2})$ if $g$ is odd;
$2q^{2g−1}+q^g+O(q^{g−1})$ if $g$is even.
