Suppose you roll a fair (cubic) die n times,
with the pay-off being the sum of the landed faces.
The expectation, variance, the (normalized) even (2r)-moment,
and odd (2r+1) moments
with asymptotics to order 2
as EXPLICIT expressions in n and r are:
/ 37 r (r - 1) r (r - 1) (r - 2) (28749 r + 20393)\
(2 r)! |1 - ------------ + -----------------------------------|
| 175 n 2 |
7 n 35 n \ 1286250 n /
---, ----, ---------------------------------------------------------------, 0
2 12 r
r! 2
This took, 0.438, seconds .