With all the recent rehashing of the relationship of variance and betting odds, some might be curious how a bet's characteristics affect its higher order moments.
Well it turns out that both skewness and kurtosis can be expressed as functions of win probability alone, without regard to odds, edge, or even bet size (this because by convention skewness and kurtosis are standardized measures). Results are as for the binomial distribution with n = 1 and trivial to prove via the definitions of each and a little simple algebra.
Skewness:
Excess Kurtosis:
Notable about both is that they're symmetric about p = ½, with only a change in sign for γ1 (negative/left skew for p > ½, positive/right skew for p < ½).
p = ½ ± 1/√12 yields a mesokurtic bet, with kurtoisis going negative as p goes to ½, where the bet is minimally kurtotic, γ2 = -2.
Well it turns out that both skewness and kurtosis can be expressed as functions of win probability alone, without regard to odds, edge, or even bet size (this because by convention skewness and kurtosis are standardized measures). Results are as for the binomial distribution with n = 1 and trivial to prove via the definitions of each and a little simple algebra.
Skewness:
γ1 = (1 - 2p)/ √p*(1-p)
Excess Kurtosis:
γ2 = 1 p + 1 (1-p) - 6
Notable about both is that they're symmetric about p = ½, with only a change in sign for γ1 (negative/left skew for p > ½, positive/right skew for p < ½).
p = ½ ± 1/√12 yields a mesokurtic bet, with kurtoisis going negative as p goes to ½, where the bet is minimally kurtotic, γ2 = -2.