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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 kurtosis of a frequency distribution is a measure of the proportion of extreme values (outliers), which appear at either end of the histogram. If the distribution is more outlier-prone than the normal distribution it is said to be leptokurtic; if less outlier-prone it is said to be platykurtic.
If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all ...
The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central and where types of departure from this include: mounded (or unimodal), U-shaped, J-shaped, reverse-J shaped and multi-modal. [1] A bimodal distribution would have two high points rather than one. The shape of a distribution is ...
One disadvantage of L-moment ratios for estimation is their typically smaller sensitivity. For instance, the Laplace distribution has a kurtosis of 6 and weak exponential tails, but a larger 4th L-moment ratio than e.g. the student-t distribution with d.f.=3, which has an infinite kurtosis and much heavier tails.
The distribution of has no closed-form expression, but can be reasonably approximated by another log-normal distribution at the right tail. [36] Its probability density function at the neighborhood of 0 has been characterized [35] and it does not resemble any log-normal distribution.
A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1895, p. 381) ′ () + + + + = ()with: =, = = +, =. According to Ord, [3] Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the density function of the normal distribution (which gives a linear function) and, secondly ...
As the logistic distribution, which can be solved analytically, is similar to the normal distribution, it can be used instead. The blue picture illustrates an example of fitting the logistic distribution to ranked October rainfalls—that are almost normally distributed—and it shows the 90% confidence belt based on the binomial distribution.