Tukey Lambda Distribution - Moments

Moments

The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for λ > −½ and is given by the formula (except when λ = 0)

\operatorname{Var} = \frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda} - \frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg).

More generally, the n-th order moment is finite when λ > −1/n and is expressed in terms of the beta function Β(x,y) (except when λ = 0) :

\mu_n = \operatorname{E} = \frac{1}{\lambda^n} \sum_{k=0}^n (-1)^k {n \choose k}\, \Beta(\lambda k+1,\, \lambda(n-k)+1 ).

Note that due to symmetry of the density function, all moments of odd orders are equal to zero.

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