How to show the least square estimator of $b$ has the minimum variance in the class $\sum a_iy_i$ Consider the regression model:
$$
y_i=bx_i+e_i,1\leq i\leq n.$$
where $x_i$'s are fixed non-zero real numbers and $e_i$'s are independent random variables with mean zero and equal variance.
$(a)$Consider estimator of the form $T=\sum_{i=1}^{n}a_iy_i$(where $a_i$'s are non random real numbers)that are unbiased for $b$.Show that the least square estimator of $b$ has the minimum variance in this class of estimators.
Minimizing $\sum e_i^{2}=\sum(y_i-bx_i)^2$ $w.r.t$ $b$,we get the least square estimate of $b$ as, $$\hat b=\frac{\sum y_ix_i}{\sum x_i^{2}}$$
Now,$var(\hat b)=\frac{\sigma^{2}}{\sum x_i^{2}}$,where $\sigma^2=var(y),$I assume.
Now,I tried to minimize $Var(T)=\sum a_i^2\sigma^{2}$ subject to $\sum a_i=1$ and I am getting $a_i=\frac{1}{n}$,i.e.,$T=\bar y$ as the $BLUE$ of $b$.I don't know where I am making the mistake.May be,my approach is not right. 
 A: Your minimum variance problem is:
$$ \begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{minimize (over $a_i$)} &  \mathrm{Var}\left(  \sum_i a_i y_i \; \middle| \; \{x_i\}\right)  \\
 \mbox{subject to} & \mathrm{E}\left[\sum_i a_iy_i \; \middle|\;\{x_i\}\right] = b 
 \end{array}
\end{equation}$$
Remember you can always substitute $y_i = b x_i + \epsilon_i$
$$ \begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{minimize (over $a_i$)} &  \mathrm{Var}\left(  \sum_i a_i (bx_i + \epsilon_i) \; \middle| \; \{x_i\}\right)  \\
 \mbox{subject to} & \mathrm{E}\left[\sum_i a_i(bx_i + \epsilon_i) \; \middle|\;\{x_i\}\right] = b 
 \end{array}
\end{equation}$$
Simplifying, you can show this is an equivalent problem to:
$$ \begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{minimize (over $a_i$)} & \sum_i a_i^2 \\
 \mbox{subject to} & \sum_i a_i x_i = 1
 \end{array}
\end{equation}$$
Which will have a nice solution! You'll see that the solution $\mathbf{a}^*$ to the above optimization problem will make your estimator $T$ equal to $\hat{b}$.
