Inverse Gaussian Distribution

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An Inverse Gaussian Distribution is a continuous probability distribution that ...

  • 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.
  • Example(s):
  • Counter-Example(s):
  • See: Wiener Process.


References

2022

  • (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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