Inverse normal distribution

The inverse normal distribution (also called the inverse Gaussian distribution or Wald distribution ) is a continuous probability distribution . It is used in generalized linear models. When examining Brownian molecular motion with drift and coefficient of scattering , the random time of the first reaching of the level is inversely normal distributed with the parameters . The inverse normal distribution belongs to the exponential family . ${\displaystyle v>0}$ ${\displaystyle \lambda >0}$ ${\displaystyle a>0}$ ${\displaystyle \left({\frac {a}{v}},{\frac {a^{2}}{\lambda ^{2}}}\right)}$

definition

Density functions of various inverse Gaussian distributions

A continuous random variable satisfies the inverse normal distribution with the parameters (event rate) and ( expected value ) if it has the probability density . ${\displaystyle X}$ ${\displaystyle \lambda >0}$ ${\displaystyle \mu >0}$ ${\displaystyle f(x)={\begin{cases}\left({\frac {\lambda }{2\pi x^{3}}}\right)^{\frac {1}{2}}e^{-{\frac {\lambda (x-\mu )^{2}}{2\mu ^{2}x}}}&x>0\\0&x\leq 0\end{cases}}}$

properties

Expected value

The inverse normal distribution has the expected value

${\displaystyle \operatorname {E} (X)=\mu }$.

Variance

The variance results analogously to

${\displaystyle \operatorname {Var} (X)={\frac {\mu ^{3}}{\lambda }}}$.

Standard deviation

This gives the standard deviation

${\displaystyle \sigma ={\sqrt {\frac {\mu ^{3}}{\lambda }}}}$

Coefficient of variation

The coefficient of variation is obtained directly from the expected value and the variance

${\displaystyle \operatorname {VarK} (X)={\sqrt {\frac {\mu }{\lambda }}}}$.

Crookedness

The skew arises too

${\displaystyle \operatorname {v} (X)=3{\sqrt {\frac {\mu }{\lambda }}}}$.

Bulge (kurtosis)

The bulge arises too

${\displaystyle \beta _{2}={\frac {15\mu }{\lambda }}+3}$.

The excess kurtosis is

${\displaystyle \gamma _{2}=\beta _{2}-3={\frac {15\mu }{\lambda }}}$.

Characteristic function

The characteristic function has the form

${\displaystyle \phi _{X}(s)=e^{{\frac {\lambda }{\mu }}\left(1-{\sqrt {1-{\frac {2\mu ^{2}is}{\lambda }}}}\right)}}$.

Moment generating function

The moment generating function of the inverse normal distribution is

${\displaystyle m_{X}(s)=e^{{\frac {\lambda }{\mu }}\left(1-{\sqrt {1-{\frac {2\mu ^{2}s}{\lambda }}}}\right)}}$.

reproducibility

Are random variables with inverse normal distribution with the parameters and , then the size is again a random variable with an inverse normal distribution, but with the parameters and . ${\displaystyle X_{1},\dots ,X_{n}}$ ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle {\frac {1}{n}}\sum \limits _{i=1}^{n}X_{i}}$ ${\displaystyle n\lambda }$ ${\displaystyle \mu }$

Web links

• Eric W. Weisstein : Inverse Gaussian Distribution . In: MathWorld (English).