Let $\displaystyle Y_{1},Y_{2},...,Y_{n}~iid~N(\mu,\sigma)$.

Q: Find the probability density function of $\displaystyle S^{2}$.

I am not sure how to approach this problem. I tried using the mgf method, but failed to find a recognizable mgf. My teacher recomened that we use the transformation method to derive the pdf, but I not sure how with this function.

Here is what he showed on the board.

Let $\displaystyle S^{2}=\frac{\sigma^{2}}{(n-1)}\frac{(n-1)S^{2}}{\sigma^{2}}$. Then, treat the preceeding quantity like $\displaystyle U=aY$ and use the tranformation method to find $\displaystyle f_{U}(u)$.

I really don't know where to go from here. Do I let $\displaystyle U=s^{2}, a=\frac{\sigma^{2}}{(n-1)},$ and $\displaystyle Y=\frac{(n-1)S^{2}}{\sigma^{2}}$?