Table 1 Types of Pearson distributions

Main Type

I

κ<0

\(f(x)=y_{0}(1+\frac {x}{a_{1}})^{m_{1}}(1-\frac {x}{a_{2}})^{m_{2}}\)

−a₁≤x≤a₂

IV

0<κ<1

\(\phantom {\dot {i}\!}f(x)=y_{0}(1+\frac {x^{2}}{a^{2}})^{-m}e^{-\nu \arctan (x/a)}\)

−∞<x<∞

VI

κ>1

\(f(x)=y_{0}(x-a)^{q_{2}}x^{-q_{1}}\phantom {\dot {i}\!}\)

a≤x<∞

Transition Type

Normal

κ=0(β₂=3)

\(\phantom {\dot {i}\!}f(x)=y_{0}e^{-x^{2}/(2\mu _{2})}\)

−∞<x<∞

II

κ=0(β₂<3)

\(f(x)=y_{0}(1-\frac {x^{2}}{a^{2}})^{m}\)

−a≤x≤a

III

κ=±∞

\(f(x)=y_{0}(1+\frac {x}{a})^{\gamma {a}}e^{-\gamma {x}}\)

−a≤x<∞

V

κ=1

f(x)=y₀x^−pe^−γ/x

0<x<∞

VII

κ=0(β₂>3)

\(f(x)=y_{0}(1+\frac {x^{2}}{a^{2}})^{-m}\)

−∞<x<∞