Choosing C so that c1Y1+c2Y2 is an Unbiased Estimator of Θ Let Y1 and Y2 be two unbiased estimators of Θ let Var(Y1)= 4Var(Y2) and the correlation coefficient between Y1 and Y2 be -.5 find constants c1 and c2 such that c1Y1 + c2Y2 is an unbiased estimator of Θ and has the smallest variance among all other linear combinations of Y1 and Y2 that are unbiased for Θ.
All I have so far is that CovY1Y2/√(4VarY2*VarY2) = CovY1Y2/2VarY2=-.5 (equation/definition of correlation coefficient)... I have no idea what to do next or even if this is a necessary step towards solving the problem. How would I know Y1+Y2 is an unbiased estimator of Θ? 
 A: In general, as mentioned in a comment, $c_1Y_1+c_2Y_2$ is an unbiased estimator of $\Theta$ if and only if $c_2=1-c_1$. For $E(c_1Y_1+c_2Y_2)=c_1E(Y_1)+c_2E(Y_2)=(c_1+c_2)\Theta$. If $\Theta\ne 0$, this is equal to $\Theta$ if and only if $c_1+c_2=1$.
Let $\sigma^2$ be the variance of $Y_2$. Then $Y_1$ has variance $4\sigma^2$. 
We find the variance of $c_1Y_1+c_2Y_2$. By a general formula you may be familiar with, we have 
$$\text{Var}(c_1Y_1+c_2Y_2)=c_1^2\text{Var}(Y_1)+c_2^2\text{Var}(Y_2)+2c_1c_2\text{Cov}(Y_1,Y_2).\tag{1}$$
You know how to find $\text{Cov}(Y_1,Y_2)$ in terms of $\sigma^2$. 
Let $c_1=t$. Then $x_2=1-t$. Write down the right-hand side of (1) in terms of $t$ and $\sigma^2$. You will get a quadratic in $t$. Choose the value of $t$ that minimizes this quadratic.  That can be done in one of the usual ways, completing the square or differentiating.
A: You have $E(Y_1) = \Theta$ and $E(Y_2) = \Theta$ from the unbiasedness assumption.
$c_1 Y_1 +c_2 Y_2$ is an unbiased estimator or the parameter if $E(c_1 Y_1 +c_2 Y_2) = \Theta$.
Hence, $c_1$ and $c_2$ must satisfy:
$E(c_1 Y_1 +c_2 Y_2) = E(Y_1)$
$E(c_1 Y_1 +c_2 Y_2) = E(Y_2)$
which by linearity of $E$ yields
$c_1 E(Y_1) +c_2 E(Y_2) = E(Y_1)$
$c_1 E(Y_1) +c_2 E(Y_2) = E(Y_2)$
where $E(Y_1) = E(Y_2) = \Theta$, hence you get only one equation which does not identify $c_1$ and $c_2$
$c_2  = (1-c_1)$
Therefore, any pair $(c_1,c_2)$ satisfying $c_1 = (1-c_2)$ will give you an unbiased estimator. Now you have to find the combination satisfying this equation that gives you the smallest variance. How do you do that?
Consider to minimize $Var(c_1 Y_1 +(1-c_1) Y_2)$ with respect to $c_1$. It should be relatively easy.
Note: Check if the expression is convex in $c_1$. If so, the argmin is characterized by the first-order condition.
