Inverse Gaussian distribution information

Inverse Gaussian
	Probability density function
	Cumulative distribution function
Notation
Parameters	;
Support
PDF
CDF	where is the standard normal (standard Gaussian) distribution c.d.f.
Mean	;
Mode
Variance	;
Skewness
Excess kurtosis
MGF
CF

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,∞).

Its probability density function is given by

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, where $\mu >0$ is the mean and $\lambda >0$ is the shape parameter.^[1]

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.

Its cumulant generating function (logarithm of the characteristic function)^{[contradictory]} is the inverse of the cumulant generating function of a Gaussian random variable.

To indicate that a random variable X is inverse Gaussian-distributed with mean μ and shape parameter λ we write $X\sim \operatorname {IG} (\mu ,\lambda )\,\!$ .

^ Cite error: The named reference Chhikara1989 was invoked but never defined (see the help page).

[Chhikara1989-1] Cite error: The named reference Chhikara1989 was invoked but never defined (see the help page).

Inverse Gaussian distribution information

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Probability distribution

Inverse Gaussian
Probability density function
Cumulative distribution function
Notation	$\operatorname {IG} \left(\mu ,\lambda \right)$
Parameters	$\mu >0$ $\lambda >0$
Support	$x\in (0,\infty )$
PDF	${\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp \left[-{\frac {\lambda (x-\mu )^{2}}{2\mu ^{2}x}}\right]$
CDF	$\Phi \left({\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}-1\right)\right)$ ${}+\exp \left({\frac {2\lambda }{\mu }}\right)\Phi \left(-{\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}+1\right)\right)$ where $\Phi$ is the standard normal (standard Gaussian) distribution c.d.f.
Mean	$\operatorname {E} [X]=\mu$ $\operatorname {E} [{\frac {1}{X}}]={\frac {1}{\mu }}+{\frac {1}{\lambda }}$
Mode	$\mu \left[\left(1+{\frac {9\mu ^{2}}{4\lambda ^{2}}}\right)^{\frac {1}{2}}-{\frac {3\mu }{2\lambda }}\right]$
Variance	$\operatorname {Var} [X]={\frac {\mu ^{3}}{\lambda }}$ $\operatorname {Var} [{\frac {1}{X}}]={\frac {1}{\mu \lambda }}+{\frac {2}{\lambda ^{2}}}$
Skewness	$3\left({\frac {\mu }{\lambda }}\right)^{1/2}$
Excess kurtosis	${\frac {15\mu }{\lambda }}$
MGF	$\exp \left[{{\frac {\lambda }{\mu }}\left(1-{\sqrt {1-{\frac {2\mu ^{2}t}{\lambda }}}}\right)}\right]$
CF	$\exp \left[{{\frac {\lambda }{\mu }}\left(1-{\sqrt {1-{\frac {2\mu ^{2}\mathrm {i} t}{\lambda }}}}\right)}\right]$