Suppose that $\displaystyle Y_{1},Y_{2},...,Y_{n}$ constitute a random sample from a normal distribution with parameters $\displaystyle \mu$ and $\displaystyle \sigma^{2}$.

Q: Show that $\displaystyle S=\sqrt{S^{2}}$ is a biased estimator of $\displaystyle \sigma$. Moreover, adjust $\displaystyle S$ to form an unbiased estimator for $\displaystyle \sigma$.

[Hint: Recall the distribution of $\displaystyle \frac{(n-1)S^{2}}{\sigma^{2}}$ and the result $\displaystyle E(Y^{a})$$\displaystyle =\frac{\beta^{a}\Gamma(\alpha\\+a)}{\Gamma(\alpha) }$].

A: Let $\displaystyle U=\frac{(n-1)S^{2}}{\sigma^{2}}$. Since $\displaystyle U$~$\displaystyle \chi^{2}(n-1)$ we have that $\displaystyle E(U)=n-1$ and $\displaystyle V(U)=2(n-1)$. Furthermore, using the result in the hint and the fact that $\displaystyle U$~$\displaystyle Gamma(\frac{n-1}{2},2)$ we have that

$\displaystyle E(\sqrt{U})$$\displaystyle =E(U^{\frac{1}{2}})$

$\displaystyle =\frac{(2)^{\frac{1}{2}}\Gamma(\frac{(n-1)}{2}+\frac{1}{2})}{\Gamma(\frac{(n-1)}{2})}$

$\displaystyle =\frac{\sqrt{2}\Gamma(\frac{n}{2})}{\Gamma(\frac{( n-1)}{2})}$.

So, $\displaystyle \sqrt{\frac{\sigma^{2}}{(n-1)}}\frac{\sqrt{2}\Gamma(\frac{n}{2})}{\Gamma(\fra c{(n-1)}{2})}$

$\displaystyle =\sigma\sqrt{\frac{2}{(n-1)}}\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{(n-1)}{2})}$.

I am not sure what do do fro the second part of the question.

Am I doing this correctly?