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:

    “I like to compare the holiday season with the way a child listens to a favorite story. The pleasure is in the familiar way the story begins, the anticipation of familiar turns it takes, the familiar moments of suspense, and the familiar climax and ending.”
    —Fred Rogers (20th century)

    “The first moments of sleep are an image of death; a hazy torpor grips our thoughts and it becomes impossible for us to determine the exact instant when the “I,” under another form, continues the task of existence.”
    —Gérard De Nerval (1808–1855)

    “To the artist is sometimes granted a sudden, transient insight which serves in this matter for experience. A flash, and where previously the brain held a dead fact, the soul grasps a living truth! At moments we are all artists.”
    —Arnold Bennett (1867–1931)