# pdf of sample variance?

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• January 20th 2010, 12:52 PM
Danneedshelp
pdf of sample variance?
Let $Y_{1},Y_{2},...,Y_{n}~iid~N(\mu,\sigma)$.

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

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

I really don't know where to go from here. Do I let $U=s^{2}, a=\frac{\sigma^{2}}{(n-1)},$ and $Y=\frac{(n-1)S^{2}}{\sigma^{2}}$?
• January 21st 2010, 08:19 PM
matheagle
Use $\sum_{i=1}^n\left({X_i-\bar X\over \sigma}\right)^2\sim\chi^2_{n-1}$