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 .
definition
A continuous random variable satisfies the inverse normal distribution with the parameters (event rate) and ( expected value ) if it has the probability density .
properties
Expected value
The inverse normal distribution has the expected value
- .
Variance
The variance results analogously to
- .
Standard deviation
This gives the standard deviation
Coefficient of variation
The coefficient of variation is obtained directly from the expected value and the variance
- .
Crookedness
The skew arises too
- .
Bulge (kurtosis)
The bulge arises too
- .
The excess kurtosis is
- .
Characteristic function
The characteristic function has the form
- .
Moment generating function
The moment generating function of the inverse normal distribution is
- .
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 .
Web links
- Eric W. Weisstein : Inverse Gaussian Distribution . In: MathWorld (English).