Inverse Gaussian Distribution

AKA: Wald Distribution.
Context:
- It can be [math]\displaystyle{ f(x;\mu,\lambda) = \sqrt\frac{\lambda}{2 \pi x^3} \exp\biggl(-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}\biggr) }[/math] for x > 0, where [math]\displaystyle{ \mu \gt 0 }[/math] is the mean and [math]\displaystyle{ \lambda \gt 0 }[/math] is the shape parameter.
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Example(s):
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Counter-Example(s):
- Normal Distribution.
- Log-Normal Distribution.
See: Wiener Process.

References

(Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Retrieved:2022-1-15.
- 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 : [math]\displaystyle{ f(x;\mu,\lambda) = \sqrt\frac{\lambda}{2 \pi x^3} \exp\biggl(-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}\biggr) }[/math] for x > 0, where [math]\displaystyle{ \mu \gt 0 }[/math] is the mean and [math]\displaystyle{ \lambda \gt 0 }[/math] is the shape parameter. 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) 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 [math]\displaystyle{ X \sim \operatorname{IG}(\mu, \lambda)\,\! }[/math] .

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