In this post: Why is sample standard deviation a biased estimator of $\sigma$?, I am having difficulty understanding some of the steps. We have :
(a) $s^2=\frac{1}{n-1}\sum_{i=1}^{\infty}(x_i-x\bar)^2$
and further down,
(b) $E(s)= \sqrt\frac{\sigma^2}{n-1} E(\sqrt\frac{s^2(n-1)}{\sigma^2})$
I understand the 2nd and 3rd lines, but how do we get from (a) to (b)? (My knowledge of maths is basic)
Ok, let's take it in steps.
$$E[s]=E\left[\sqrt{s^2}\right]$$
Now introduce some fractions that are equal to 1 and don't make a difference $$E\left[\sqrt{s^2}\right]=E\left[\sqrt\frac{n-1}{n-1}\times\sqrt\frac{\sigma^2}{\sigma^2}\times\sqrt{s^2}\right]$$ Now, the $\sqrt{\sigma^2}$ in the numerator and the $\sqrt{(n-1)}$ in the denominator are constants and can come out of the expectation $$E\left[\sqrt\frac{n-1}{n-1}\times\sqrt\frac{\sigma^2}{\sigma^2} \times\sqrt{s^2}\right]=\sqrt{\frac{\sigma^2}{n-1}}E\left[\sqrt\frac{n-1}{1}\times\sqrt\frac{1}{\sigma^2}\times\sqrt{s^2}\right]=\sqrt{\frac{\sigma^2}{n-1}}E\left[\sqrt{\frac{n-1}{\sigma^2}\times s^2}\right]$$ which was to be shown.
