Inverse Gaussian Distribution

Introduction

The inverse Gaussian distribution (IG) (also known as Wald distribution) is a two-parameter continuous distribution given by its density function

$$f(x;\mu ,\lambda ) = \sqrt{ \frac{\lambda } {2\pi }}{x}^{-3/2}\exp \left \{- \frac{\lambda } {2{\mu }^{2}x}{(x - \mu )}^{2}\right \},\ \ x > 0.$$

The parameter μ > 0 is the mean and λ > 0 is the shape parameter. For a random variable (r.v.) X with inverse Gaussian distribution we write X ∼ IG(μ, λ).

The inverse Gaussian distribution describes the distribution of the time a Brownian motion (see Brownian Motion and Diffusions) with positive drift takes to reach a given positive level. To be precise, let ${X}_{t}\,=\,\nu t + \sigma {W}_{t}$ be a Brownian motion with drift ν > 0 (here W _t is the standard Brownian motion). Let T _a be the first passage time for a fixed level a > 0 by X _t . Then T _a has inverse Gaussian distribution, ${T}_{a} \sim IG\left (\frac{a} {\nu }, \frac{{a}^{2}} {{\sigma }^{2}} \right ).$

The inverse...