= Inverse Gaussian sum closure
{title2=$S_n\sim IG(\phi,n^2\lambda)$}
In the parametrization with density proportional to $y^{-3/2}\exp[-(\lambda/y+\phi y)/2]$, the <Inverse Gaussian distribution> has <cumulant-generating function> $K(t)=\sqrt{\lambda\phi}-\sqrt{\lambda(\phi-2t)}$. For $n$ independent copies, the average has this same family with parameters $(n\phi,n\lambda)$, while the sum has parameters $(\phi,n^2\lambda)$. These assertions follow by multiplying <moment-generating functions> and scaling the density. The more usual mean parameter is $\mu=\sqrt{\lambda/\phi}$, so the sum has mean $n\mu$ and shape $n^2\lambda$.
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