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.

Read more about this topic:  Tukey Lambda Distribution

Famous quotes containing the word moments:

    “Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
    —Jane Heap (c. 1880–1964)

    “The essence of the modern state is that the universal be bound up with the complete freedom of its particular members and with private well-being, that thus the interests of family and civil society must concentrate themselves on the state.... It is only when both these moments subsist in their strength that the state can be regarded as articulated and genuinely organized.”
    —Georg Wilhelm Friedrich Hegel (1770–1831)

    “What moments divine, what rapture serene.”
    —Cole Porter (1893–1964)