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The standard measure of a distribution's kurtosis, originating with Karl Pearson, [1] is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; [ 2 ] hence, the sometimes-seen characterization of kurtosis as " peakedness " is incorrect.
The fourth central moment is a measure of the heaviness of the tail of the distribution. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is 3σ 4.
Therefore, all of the cokurtosis terms of this distribution with this nonlinear correlation are smaller than what would have been expected from a bivariate normal distribution with ρ=0.818. Note that although X and Y are individually standard normally distributed, the distribution of the sum X+Y is platykurtic. The standard deviation of the sum is
The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids. Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0
As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random ...
As odd cumulants of a uniform distribution are zero; only even moments are affected. The second and fourth cumulants of the uniform distribution on (−0.5c, 0.5c) are respectively, c 2 /12 and −c 4 /120. The correction to moments can be derived from the relation between cumulants and moments.
The exponentially modified Gaussian distribution, a convolution of a normal distribution with an exponential distribution, and the Gaussian minus exponential distribution, a convolution of a normal distribution with the negative of an exponential distribution. The expectile distribution, which nests the Gaussian distribution in the symmetric case.
The ordinary cumulants of degree higher than 2 of the normal distribution are zero. The free cumulants of degree higher than 2 of the Wigner semicircle distribution are zero. [22] This is one respect in which the role of the Wigner distribution in free probability theory is analogous to that of the normal distribution in conventional ...