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:

    “As the tragic writer rids us of what is petty and ignoble in our nature, so also the humorist rids us of what is cautious, calculating, and priggish—about half of our social conscience, indeed. Both of them permit us, in blessed moments of revelation, to soar above the common level of our lives.”
    —Robertson Davies (b. 1913)

    “There are moments when you feel free, moments when you have energy, moments when you have hope, but you can’t rely on any of these things to see you through. Circumstances do that.”
    —Anita Brookner (b. 1938)

    “There are thoughts which are prayers. There are moments when, whatever the posture of the body, the soul is on its knees.”
    —Victor Hugo (1802–1885)