The question is:
If $X$ is a binomial random variable with parameters $n$ and $\theta$, show that $n \left( \frac{X}{n} \right)\left( 1-\frac{X}{n} \right)$ is a biased estimator of the variance of $X$.
So my way to tackle this question would be:
$$ \mathbf{E}\left[ n \left( \frac{X}{n} \right)\left( 1-\frac{X}{n} \right) \right] = \mathbf{E}\left[ X - \frac{X^2}{n} \right] = \mathbf{E}[X] - \frac{1}{n} \mathbf{E}[X^2] $$
After that, I just plug in the given values of $\mathbf{E}[X]$ and $\mathbf{E}[X^2]$ for the binomial distribution, but my teacher say that is wrong saying that I should end up with $n \mathbf{E}[X] - \mathbf{E}[X^2]$. I don't see a way how there is another $n$? unless I'm missing something.
Help would be much appreciated. Thanks