In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on ( 0 , ∞ ) {\displaystyle (0,\infty )} . Its probability density function is given by
f ( x ; μ , λ ) = λ 2 π x 3 exp ( − λ ( x − μ ) 2 2 μ 2 x ) {\displaystyle f(x;\mu ,\lambda )={\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp {\biggl (}-{\frac {\lambda (x-\mu )^{2}}{2\mu ^{2}x}}{\biggr )}}
for x > 0 {\displaystyle x>0} , where μ > 0 {\displaystyle \mu >0} is the mean and λ > 0 {\displaystyle \lambda >0} is a shape parameter. Either μ {\displaystyle \mu } or λ {\displaystyle \lambda } (or more generally any combination of the form μ p λ 1 − p {\displaystyle \mu ^{p}\lambda ^{1-p}} for any real p {\displaystyle p} ) can serve as a scale parameter, so a proper (i.e., unscaled) shape parameter would be any non-zero power of φ = λ / μ {\displaystyle \varphi =\lambda /\mu } : Tweedie proposed to use the ( μ , φ ) {\displaystyle (\mu ,\varphi )} and ( φ , λ ) {\displaystyle (\varphi ,\lambda )} parametrizations in addition to the standard ( μ , λ ) {\displaystyle (\mu ,\lambda )} parametrization (“Each of these forms is convenient or suggestive for some purpose.”), and later on uses exclusively the ( φ , λ ) {\displaystyle (\varphi ,\lambda )} parametrization. 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 fixed positive level. The relationship between the Gaussian and inverse Gaussian distributions is thus the same as the relationship between the binomial (number of successes for a fixed number of Bernoulli trials) and negative binomial (number of Bernoulli trials for a fixed number of successes) distributions. The y-axis reflections of the cumulant generating functions of the Gaussian and inverse Gaussian distributions are inverse of each other (i.e., the graphs of the two cumulant generating functions are reflections of each other across the line y = − x {\displaystyle y=-x} ), a property that is also shared between the binomial and negative binomial distributions (after dividing their cumulant generating functions by their respective fixed parameter). To indicate that a random variable X {\displaystyle X} is inverse Gaussian-distributed with mean μ {\displaystyle \mu } and shape parameter λ {\displaystyle \lambda } we write X ∼ IG ( μ , λ ) {\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )} .
Properties
Single parameter form The probability density function (pdf) of the inverse Gaussian distribution has a single parameter form given by
f ( x ; μ , μ 2 ) = μ 2 π x 3 exp ( − ( x − μ ) 2 2 x ) . {\displaystyle f(x;\mu ,\mu ^{2})={\frac {\mu }{\sqrt {2\pi x^{3}}}}\exp {\biggl (}-{\frac {(x-\mu )^{2}}{2x}}{\biggr )}.}
In this form, the mean and variance of the distribution are equal, E [ X ] = Var ( X ) {\displaystyle \mathbb {E} [X]=\operatorname {Var} (X)} . Also, the cumulative distribution function (cdf) of the single parameter inverse Gaussian distribution is related to the standard normal distribution by
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