if I have $\displaystyle \frac {x}{n} + x - \frac{n}{n-1}$ how can I transform it into $\displaystyle x + nx - \frac{n}{n-1}$?

I can't seem to do it without changing the n/(n-1) term into n^2/(n-1)

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- Nov 5th 2013, 12:54 AMkingsolomonsgravetransforming one equation into another
if I have $\displaystyle \frac {x}{n} + x - \frac{n}{n-1}$ how can I transform it into $\displaystyle x + nx - \frac{n}{n-1}$?

I can't seem to do it without changing the n/(n-1) term into n^2/(n-1) - Nov 5th 2013, 04:35 AMSlipEternalRe: transforming one equation into another
Please give the whole problem. The expression $\displaystyle \dfrac{x}{n}+x - \dfrac{n}{n-1}$ is a different expression from $\displaystyle x+nx - \dfrac{n}{n-1}$. I would need the full context to be able to help further.

If $\displaystyle \dfrac{x}{n} + x - \dfrac{n}{n-1} = 0$ then $\displaystyle x + nx - \dfrac{n^2}{n-1} = 0$ as you suggested. Or, $\displaystyle \dfrac{x}{n} + x - \dfrac{n}{n-1} = \dfrac{1}{n}(x + nx) - \dfrac{n}{n-1}$. Without more information, I don't know how to help you. - Nov 5th 2013, 05:12 AMkingsolomonsgraveRe: transforming one equation into another
sorry about that. It's a statistics problem that becomes an algebra problem.

I have as an estimator of the variance of the population$\displaystyle \frac{\sum(y_i - \bar{y})^2}{n} - \frac{n}{n-1} + \sum(y_i - \bar{y})^2$ and I'm trying to find out if it's biased or not, meaning if I take the expectation whether or not the expression remains the same. When I take the expectation I get

$\displaystyle \sum(y_i - \bar{y})^2 - \frac{n}{n-1} + n \sum(y_i - \bar{y})^2$ (I hope that's right, if not the whole thing here is wrong) and the question is how would I make this unbiased, which I interpret to mean that I have to find some way of algebraically making the second statement equivalent to the first. In the above problem I let $\displaystyle E( \sum (y_i - \bar{y})^2 )$= VARIANCE(Y) = x so as to simplify the algebra.

It could be that my statistics up to this point are wrong and perhaps this is why the algebra is not working. - Nov 5th 2013, 05:58 AMSlipEternalRe: transforming one equation into another
Ahh, my knowledge of statistics is rather limited. But, would this help with the algebra? Rewriting your original expression, you have

$\displaystyle \dfrac{\sum{(y_i-\overline{y})^2}}{n} - \dfrac{n}{n-1} + \sum{(y_i - \overline{y})^2} = \dfrac{n+1}{n}\sum{(y_i - \overline{y})^2} - \dfrac{n}{n-1}$

If you find the expected value of the RHS expression, do you get $\displaystyle (n+1)\sum(y_i-\overline{y})^2 - \dfrac{n}{n-1}$? If so, then your algebra looks correct to me, and maybe the estimator is biased.