Chi-squared transform of an inverse Gaussian variable
= Chi-squared transform of an inverse Gaussian variable
{title2=$\lambda(X-\mu)^2/(\mu^2X)\sim\chi_1^2$}
For $X\sim IG(\mu,\lambda)$, $h(X)=\lambda(X-\mu)^2/(\mu^2X)$ has the chi-squared-one law. Its two inverse branches contribute weights $\mu/(\mu+x_-)$ and $\mu/(\mu+x_+)$ to the transformed density, and reciprocal roots make those weights sum to one.