# Thread: transforming one equation into another

1. ## transforming one equation into another

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

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

2. ## Re: transforming one equation into another

Please give the whole problem. The expression $\dfrac{x}{n}+x - \dfrac{n}{n-1}$ is a different expression from $x+nx - \dfrac{n}{n-1}$. I would need the full context to be able to help further.

If $\dfrac{x}{n} + x - \dfrac{n}{n-1} = 0$ then $x + nx - \dfrac{n^2}{n-1} = 0$ as you suggested. Or, $\dfrac{x}{n} + x - \dfrac{n}{n-1} = \dfrac{1}{n}(x + nx) - \dfrac{n}{n-1}$. Without more information, I don't know how to help you.

3. ## Re: 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 $\frac{\sum(y_i - \bar{y})^2}{n} - \frac{n}{n-1} + \sum(y_i - \bar{y})^2$ 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

$\sum(y_i - \bar{y})^2 - \frac{n}{n-1} + n \sum(y_i - \bar{y})^2$ (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 $E( \sum (y_i - \bar{y})^2 )$= 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.

4. ## Re: 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

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

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