How to Prove Unbiased Estimator I'm unsure of how to convince myself that 
$$\hat{\beta} = \frac{\sum X_i Y_i}{\sum X_i^2}$$
is an unbiased estimator when the regression model
$$Y_i = \beta X_i + \epsilon_i$$
follows basic OLS assumptions. To show this is unbiased, we need to show that $\text{E}( \hat{\beta} ) = \beta$.
My hunch is that the $X_i$ and $X_i^2$ will cancel out to give $\frac{Y_i}{X_i}$ (which is what I think $\beta$ equals?, but I'm not sure how to show it with the expectation). I get stuck with the $\text{E}(\sum X_i^2)$.  My understanding is that this should equal $\text{Var}(X) \cdot \bar{X}^2$ This only gives $\text{E}(\hat{\beta}) = \beta$ if $\text{Var}(X) = 1$.  
 A: The key is that we are conditioning on the predictors $\{x_1, x_2, \ldots, x_n \}$ so they're viewed as constants.  Once we realize this it becomes very straightforward:
\begin{align}
\text{E} \left ( \frac{\sum_i x_i Y_i}{\sum_j x_j^2} \right ) &= \frac{ \sum_i x_i \text{E}(Y_i)}{\sum_j x_j^2} \\
&= \frac{ \beta \sum_i x_i^2}{\sum_j x_j^2} \\
&= \beta 
\end{align}
where we've used the fact that $\text{E}(Y_i) = \text{E}(\beta x_i + \epsilon_i) = \beta x_i$ since $\text{E}(\epsilon_i) = 0$.
A: The first thing to point out, is that your target equation
$$ \hat \beta = \beta $$
cannot be correct.  The left hand side is a random variable, and the right hand side is a constant, so there is no hope to prove them equal, no matter how ingenious the algebra.
What you really want for unbiasedness is to show this
$$ E[\hat \beta \mid X] = \beta $$
Keeping in mind that conditioning allows you to treat $X$ as a constant, you should be able to get it from there.
A: Taking a look given below might solve your query.
https://www.youtube.com/watch?v=PriultFg8Qo
