
pdf of sample variance?
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 $\displaystyle S^{2}=\frac{\sigma^{2}}{(n1)}\frac{(n1)S^{2}}{\sigma^{2}}$. 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 $\displaystyle U=s^{2}, a=\frac{\sigma^{2}}{(n1)},$ and $\displaystyle Y=\frac{(n1)S^{2}}{\sigma^{2}}$?

Use $\displaystyle \sum_{i=1}^n\left({X_i\bar X\over \sigma}\right)^2\sim\chi^2_{n1}$