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 This took, 0.321, seconds .