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:

    “Parenting is not logical. If it were, we would never have to read a book, never need a family therapist, and never feel the urge to call a close friend late at night for support after a particularly trying bedtime scene. . . . We have moments of logic, but life is run by a much larger force. Life is filled with disagreement, opposition, illusion, irrational thinking, miracle, meaning, surprise, and wonder.”
    —Jeanne Elium (20th century)

    “The amelioration of the world cannot be achieved by sacrifices in moments of crisis; it depends on the efforts made and constantly repeated during the humdrum, uninspiring periods, which separate one crisis from another, and of which normal lives mainly consist.”
    —Aldous Huxley (1894–1963)

    “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)