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| Keywords: Pearson type VII distribution PDF.png Image summary Probability density function of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the blue curve shows the density with kurtosis equal to 2; the black curve shows the limiting normal density with kurtosis identically zero Background The Pearson type VII family of probability densities is a special case of the type IV family restricted to symmetric densities The probability density function is given by <math>f x; a 2 m \frac \Gamma m \Gamma m-1/2 \ \sqrt a 2\ \pi \left 1+\frac x 2 a 2 \right -m \ </math> All densities in this family are symmetric with zero mean Setting <math>a 2 2m-3</math> makes the variance equal to unity Then the only free parameter is m which controls the fourth moment and cumulant and hence the kurtosis One can reparameterize with <math>m 5/2 + 3/k</math> where k is the kurtosis to obtain a one-parameter leptokurtic family with zero mean unit variance zero skew and arbitrary positive kurtosis k In the limit as <math>k \to \infty</math> one obtains the density <math>g x 3 \left 2 + x 2\right -5/2 \ </math> shown as the red curve In the other direction as <math>k \to 0</math> one obtains the standard normal density as the limiting distribution shown as the black curve n the Pearson type VII log-pdf log_p7 x a2 m lgamma m - lgamma m-0 5 - m log 1+x x/a2 - 0 5 log a2 pi the Pearson type VII log-pdf with unit variance and kurtosis k f x k log_p7 x 2+6 0/k 2 5+3 0/k the standard normal log-pdf with unit variance and kurtosis 0 n x -0 5 x x + log 2 pi the limit of the Pearson type VII log-pdf for k -> infinity g x -2 5 log 2 + x x + log 3 set samples 1001 set grid set xrange -10 4 10 4 set xtics 1 set yrange -0 02 0 57 set ytics 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 39894 0 45553 0 5 0 53033 set terminal postscript enhanced color solid lw 1 Times-Roman 17 set output Pearson_type_VII_distribution_PDF ps plot exp g x lt 1 lw 2 notitle \ exp f x 2 lt 3 lw 2 notitle \ exp n x lt 7 lw 2 notitle set output Pearson_type_VII_distribution_log-PDF ps set xrange -10 4 10 4 set xtics 1 set yrange -55 2 set ytics 5 plot g x lt 1 lw 2 notitle \ f x 2 lt 3 lw 2 notitle \ f x 1 lt 9 lw 1 notitle \ f x 0 5 lt 9 lw 1 notitle \ f x 0 25 lt 9 lw 1 notitle \ f x 0 125 lt 9 lw 1 notitle \ f x 0 0625 lt 9 lw 1 notitle \ n x lt 7 lw 2 notitle GPL graph See also Image Pearson type VII distribution log-PDF png probability distributions graph | ||||
