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The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a ...
The cumulative distribution function (shown as F(x)) gives the p values as a function of the q values. The quantile function does the opposite: it gives the q values as a function of the p values. Note that the portion of F ( x ) in red is a horizontal line segment.
The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively. [7] Specifically, an inverse Gaussian distribution of the form
The cumulative distribution function of the reciprocal, within the same range, is G ( y ) = b − y − 1 b − a . {\displaystyle G(y)={\frac {b-y^{-1}}{b-a}}.} For example, if X is uniformly distributed on the interval (0,1), then Y = 1 / X has density g ( y ) = y − 2 {\displaystyle g(y)=y^{-2}} and cumulative distribution function G ( y ...
The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: [9] if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and ,, respectively, then + is NIG-distributed with parameters ,, + and +.
Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if is near zero), obtained by setting =, the probability density function is given by f Y ( y ; θ ) = 2 θ π exp ( − y 2 θ 2 π ) y ≥ 0 , {\displaystyle f_{Y}(y;\theta )={\frac {2\theta }{\pi }}\exp \left(-{\frac {y^{2}\theta ^{2}}{\pi ...
The noncentral F-distribution is implemented in the R language (e.g., pf function), in MATLAB (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) in Mathematica (NoncentralFRatioDistribution function), in NumPy (random.noncentral_f), and in Boost C++ Libraries.
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding: