The Expected Embedding Dimension, type and weight of a Numerical Semigroup

We study statistical properties of numerical semigroups of genus $g$ as $g$ goes to infinity. More specifically, we answer a question of Eliahou by showing that as $g$ goes to infinity, the proportion of numerical semigroups of genus $g$ with embedding dimension close to $g/\sqrt{5}$ approaches $1$. We prove similar results for the type and weight of a numerical semigroup of genus $g$.