Table 1.

Types of Pearson distributions

Type	κ-Criterion	Density function	Domain
Main Type
I	κ<0	$f\left(x\right)=y_0{(1+\frac{x}{a_1})}^{m_1}{(1-\frac{x}{a_2})}^{m_2}$	−a1≤x≤a2
IV	0<κ<1	$f\left(x\right)=y_0{(1+\frac{x^2}{a^2})}^{-m}{e}^{-\nu arctan(x/a)}$	−∞<x<∞
VI	κ>1	$f\left(x\right)=y_0{(x-a)}^{q_2}{x}^{-q_1}$	a≤x<∞
Transition Type
Normal	κ=0(β2=3)	$f\left(x\right)=y_0{e}^{-x^2/\left(2{\mu }_2\right)}$	−∞<x<∞
II	κ=0(β2<3)	$f\left(x\right)=y_0{(1-\frac{x^2}{a^2})}^m$	−a≤x≤a
III	κ=±∞	$f\left(x\right)=y_0{(1+\frac{x}{a})}^{\mathrm{\gamma a}}{e}^{-\mathrm{\gamma x}}$	−a≤x<∞
V	κ=1	f(x)=y0x−pe−γ/x	0<x<∞
VII	κ=0(β2>3)	$f\left(x\right)=y_0{(1+\frac{x^2}{a^2})}^{-m}$	−∞<x<∞