# Solving a variance with two estimators and known covariance

I want to solve the following variance of an estimator but after many trials I still didn't succeed...

$V(a\hat{\theta}_1 + (1-a)\hat{\theta}_2)$

We have that:

• $V(\hat θ_1) = σ_1^2$
• $V(\hat θ_2) = σ_2^2$
• $\operatorname{Cov}(\hat θ_1, \hat θ_2) = c ≠ 0$

Any idea?

• When you extract from the covariance $a$ and $(1-a)$, is that a property of covariance? I don't get that step... Thanks for you answer by the way! :) – JaviOverflow Aug 9 '17 at 18:14
• @JaviOverflow : Another standard exercise is to use the definition of covariance to show that $$\operatorname{cov}(aX, bY) = ab\operatorname{cov}(X,Y).$$ That's when $a$ and $b$ are real numbers. If they are matrices, then one gets $a \operatorname{cov}(X,Y) b^T. \qquad$ – Michael Hardy Aug 9 '17 at 18:17