Scale transformation of gamma variable

Question

we know that $(n −1)S^2/\sigma^2 ∼ \chi^2(n−1) = gamma((n − 1)/2, 2)$.

Now I want to know the distribution of $S^2$. How to do that?

Answer 1

If $\frac{n −1}{\sigma^2}S^2 \sim \mathrm{gamma}\left(\frac{n − 1}2, 2\right)$ using a shape-scale parametrisation

then $S^2 \sim \mathrm{gamma}\left(\frac{n − 1}2, \frac{2\sigma^2}{n-1}\right)$ by rescaling