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:

    “All the strong agonized men
    Wear the hard clothes of war,
    Try to remember what they are fighting for.
    But in dark weeping helpless moments of peace
    Women and poets believe and resist forever.”
    —Muriel Rukeyser (1913–1980)

    “Athletes have studied how to leap and how to survive the leap some of the time and return to the ground. They don’t always do it well. But they are our philosophers of actual moments and the body and soul in them, and of our manoeuvres in our emergencies and longings.”
    —Harold Brodkey (b. 1930)

    “Science with its retorts would have put me to sleep; it was the opportunity to be ignorant that I improved. It suggested to me that there was something to be seen if one had eyes. It made a believer of me more than before. I believed that the woods were not tenantless, but choke-full of honest spirits as good as myself any day,—not an empty chamber, in which chemistry was left to work alone, but an inhabited house,—and for a few moments I enjoyed fellowship with them.”
    —Henry David Thoreau (1817–1862)