Just a tiny addition: Yes, N log N is the average time, but the distribution is heavily long-tailed, the variance is quite high, so in many instances it might take quite some time till every item has been visited (in contrast to merely most items).

The keyword to look up more details is "coupon collector's problem".

You can also cover every one of the points "with high probability" in O(N log N) time (meaning: the chance you missed any point is at most 1/p(N) for a polynomial p, with the constant in the big-O depending on p.)