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:

    “Within the circuit of this plodding life
    There enter moments of an azure hue,
    Untarnished fair as is the violet
    Or anemone, when the spring strews them
    By some meandering rivulet, which make
    The best philosophy untrue that aims
    But to console man for his grievances.
    I have remembered when the winter came,”
    —Henry David Thoreau (1817–1862)

    “Insults from an adolescent daughter are more painful, because they are seen as coming not from a child who lashes out impulsively, who has moments of intense anger and of negative feelings which are not integrated into that large body of responses, impressions and emotions we call ‘our feelings for someone,’ but instead they are coming from someone who is seen to know what she does.”
    —Terri Apter (20th century)

    “For these are moments only, moments of insight,
    And there are reaches to be attained,
    A last level of anxiety that melts
    In becoming, like miles under the pilgrim’s feet.”
    —John Ashbery (b. 1927)