  Consider a set $P$ of $n$ points picked uniformly and independently from
$[0,1]^d$ for a constant dimension $d$ -- such a point set is extremely well
behaved in many aspects. For example, for a fixed $r \in [0,1]$, we prove a new
concentration result on the number of pairs of points of $P$ at a distance at
most $r$ -- we show that this number lies in an interval that contains only
$O(n \log n)$ numbers.
  We also present simple linear time algorithms to construct the Delaunay
triangulation, Euclidean MST, and the convex hull of the points of $P$. The MST
algorithm is an interesting divide-and-conquer algorithm which might be of
independent interest. We also provide a new proof that the expected complexity
of the Delaunay triangulation of $P$ is linear -- the new proof is simpler and
more direct, and might be of independent interest. Finally, we present a simple
$\tilde{O}(n^{4/3})$ time algorithm for the distance selection problem for
$d=2$.
