The inverse Gaussian distribution $IG(\mu,\lambda)$ is associated with the density
$$f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\,\exp\left\{-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right\}\qquad \lambda,\mu,x>0$$
In [Schuster (1968)][1], the following connection with the $\chi^2(1)$ distribution is made: if $X\sim IG(\mu,\lambda)$ then$$Z=\frac{\lambda(X-\mu)^2}{2\mu^2X}\sim\chi^2(1)$$
When looking at the proof

[![enter image description here][2]][2]

I cannot fill the gap between the definition of $Z$ [as a one-to-one transform of $Y$] and the "immediate" conclusion that it is a $\chi^2(1)$ variate. The [1978 review by Folks and Chhikara][3] does not provide further enlightenment.


  [1]: https://amstat.tandfonline.com/doi/pdf/10.1080/01621459.1968.10480942?needAccess=true
  [2]: https://i.sstatic.net/oULam.png
  [3]: https://www.jstor.org/stable/2984691?seq=3#metadata_info_tab_contents