Simple linear regression model

The OLS estimator for the simple linear regression model can be written as

$\displaystyle

b_2 = \beta_2 + \Sigma w_te_t$ where $\displaystyle w_t = \frac{(x_t - \overline{x})}{\Sigma(x_t - \overline{x})^2}

$

In the case for a sample size of three observations (T=3), this estimator can be written as:

$\displaystyle

b_2 = \beta_2 + w_1e_1 + w_2e_2 + w_3e_3$ where $\displaystyle w_t = \frac{(x_t - \overline{x})}{\Sigma(x_t - \overline{x})^2}, t = 1,2,3

$

Assume that the full set of assumption of the SLRM hold.

Without using summation notation, use the fact that $\displaystyle E[b_2]=\beta_2$ to find $\displaystyle V[b_2]$.

Explain how the assumptions of the model are used to arrive at your answer.

i started by equating $\displaystyle b_2 = E[ (b_2 - E(b_2) )^2]$ (definition of variance),

then using known results to

$\displaystyle = E(b_2) - [E(b_2)]^2$

would greatly appreciate if anyone could help me derive $\displaystyle V(b_2)$ without using any summation notation.

i suppose the end resut of $\displaystyle V(b_2)$ [using a proof that uses summation notation) would be

$\displaystyle \frac{(\sigma^2)}{\Sigma(x_t - \overline{x})^2}$

and,

$\displaystyle E[(e_t)^2] = \sigma^2$

and $\displaystyle w_t = \frac{(x_t - \overline{x})}{\Sigma(x_t - \overline{x})^2}$