I need to check if an estimator $\hat\beta = \frac{1}{n}\sum\limits_{i=1}^{n} \frac{Y_i-\bar{Y}}{X_i-\bar{X}}$ of regression $ Y_i = \alpha +\beta X_i + \epsilon_i, i = 1,...n $ is unbiased.
My idea is to check if $E\left[\hat{\beta}\right] = \beta$, so
$$E[\hat{\beta}] = E\left[\frac{1}{n}\sum_{i=1}^n \frac{Y_i-\bar{Y}}{X_i-\bar{X}}\right] = \frac{1}{n} \sum_{i=1}^n E\left[\frac{Y_i-\bar{Y}}{X_i-\bar{X}}\right] = \frac{1}{n} \sum_{i=1}^n E\left[\frac{\alpha +\beta X_i + \epsilon_i-\bar{Y}}{X_i-\bar{X}}\right]$$
At that stage I have no idea how to continue.