page_title: Beta distribution
section_title: Kurtosis bounded by the square of the skewness
hierarchical_section_title: Beta distribution / Properties / Relationships between statistical measures / Kurtosis bounded by the square of the skewness
caption_attribution_description: English: (alpha and beta) Parameter estimates vs. excess Kurtosis and (squared) Skewness Beta distribution - J. Rodal
context_page_description: In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.
  The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
  In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The beta distribution is a suitable model for the random behavior of percentages and proportions.
  The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.
context_section_description: As remarked by Feller, in the Pearson system the beta probability density appears as type I (any difference between the beta distribution and Pearson's type I distribution is only superficial and it makes no difference for the following discussion regarding the relationship between kurtosis and skewness). Karl Pearson showed, in Plate 1 of his paper   published in 1916,  a graph with the kurtosis as the vertical axis (ordinate) and the square of the skewness as the horizontal axis (abscissa), in which a number of distributions were displayed.  The region occupied by the beta distribution is bounded by the following two lines in the (skewness²,kurtosis) plane, or the (skewness²,excess kurtosis) plane:
  or, equivalently,
  (At a time when there were no powerful digital computers), Karl Pearson accurately computed further boundaries, for example, separating the "U-shaped" from the "J-shaped" distributions. The lower boundary line (excess kurtosis + 2 − skewness² = 0) is produced by skewed "U-shaped" beta distributions with both values of shape parameters α and β close to zero.  The upper boundary line (excess kurtosis − (3/2) skewness² = 0) is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter.  Karl Pearson showed  that this upper boundary line (excess kurtosis − (3/2) skewness² = 0) is also the intersection with Pearson's distribution III, which has unlimited support in one direction (towards positive infinity), and can be bell-shaped or J-shaped. His son, Egon Pearson, showed  that the region (in the kurtosis/squared-skewness plane) occupied by the beta distribution (equivalently, Pearson's distribution I) as it approaches this boundary (excess kurtosis − (3/2) skewness² = 0) is shared with the noncentral chi-squared distribution.  Karl Pearson (Pearson 1895, pp. 357, 360, 373–376) also showed that the gamma distribution is a Pearson type III distribution. Hence this boundary line for Pearson's type III distribution is known as the gamma line. (This can be shown from the fact that the excess kurtosis of the gamma distribution is 6/k and the square of the skewness is 4/k, hence (excess kurtosis − (3/2) skewness² = 0) is identically satisfied by the gamma distribution regardless of the value of the parameter "k"). Pearson later noted that the chi-squared distribution is a special case of Pearson's type III and also shares this boundary line (as it is apparent from the fact that for the chi-squared distribution the excess kurtosis is 12/k and the square of the skewness is 8/k, hence (excess kurtosis − (3/2) skewness² = 0) is identically satisfied regardless of the value of the parameter "k"). This is to be expected, since the chi-squared distribution X ~ χ²(k) is a special case of the gamma distribution, with parametrization X ~ Γ(k/2, 1/2) where k is a positive integer that specifies the "number of degrees of freedom" of the chi-squared distribution.
  An example of a beta distribution near the upper boundary (excess kurtosis − (3/2) skewness² = 0) is given by α = 0.1, β = 1000, for which the ratio (excess kurtosis)/(skewness²) = 1.49835 approaches the upper limit of 1.5 from below. An example of a beta distribution near the lower boundary (excess kurtosis + 2 − skewness² = 0) is given by α= 0.0001, β = 0.1, for which values the expression (excess kurtosis + 2)/(skewness²) = 1.01621 approaches the lower limit of 1 from above. In the infinitesimal limit for both α and β approaching zero symmetrically, the excess kurtosis reaches its minimum value at −2.  This minimum value occurs at the point at which the lower boundary line intersects the vertical axis (ordinate). (However, in Pearson's original chart, the ordinate is kurtosis, instead of excess kurtosis, and it increases downwards rather than upwards).
  Values for the skewness and excess kurtosis below the lower boundary (excess kurtosis + 2 − skewness² = 0) cannot occur for any distribution, and hence Karl Pearson appropriately called the region below this boundary the "impossible region." The
caption: Beta distribution α and β parameters vs. excess Kurtosis and squared Skewness
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