If you toss a fair coin n times, winning a dollar with Heads and losing a dollar with Tails
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:
/ r (r - 1) r (r - 1) (r - 2) (5 r + 1)\
(2 r)! |1 - --------- + ---------------------------|
| 3 n 2 |
\ 90 n /
0, n, ----------------------------------------------------, 0
r
r! 2
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