Inverse Gaussian Distribution

Definition

$X\sim IG(\mu ,\lambda )$ where $\mu >0$ is the mean and $\lambda >0$ is the scale parameter.

Properties

PDF

$f(x)=\sqrt{\frac{\lambda }{2\pi x^3}}\exp \left(-\frac{\lambda (x-\mu {)}^2}{2\mu x}\right)\in {\mathbb{N}}_0$

Mean

$E(X)=\mu$

Variance

$\mathrm{Var}(X)=\frac{{\mu }^3}{\lambda }$

MGF

$M(t)=\exp \left[\frac{\lambda }{\mu }\left(1-\sqrt{1-\frac{2{\mu }^{2}t}{\lambda }}\right)\right]$

Summation

Let $X_i\sim IG({\mu }_0{w}_i,{\lambda }_0{w}_i^2)$, and $X_i$‘s are independent, then $\sum _{i=1}^n{X}_i\sim IG\left({\mu }_0\sum _{i=1}^n{w}_i,{\lambda }_0{\left(\sum _{i=1}^n{w}_i\right)}^2\right)$