# $Y_1,Y_2,Y_3$ are uncorrelated rvs such that $E(Y_1)=\beta_1+\beta_2$,$E(Y_2)=2\beta_1$ and $E(Y_3)=\beta_1-\beta_2$

Let $$Y_1,Y_2,Y_3$$ be uncorrelated random variables with common variance $$\sigma^2 > 0$$ such that $$E(Y_1)=\beta_1+\beta_2$$,$$E(Y_2)=2\beta_1$$ and $$E(Y_3)=\beta_1-\beta_2$$ where $$\beta_1$$ and $$\beta_2$$ are unknown parameters.We need to find the residual (error) sum of squares under the above linear model.

Can anyone give an idea how to approach this? I can't think of an approach.

The residual sum of squares is $$(y_1-(b_1+b_2))^2+(y_2-2 b_1)^2+(y_3-(b_1-b_2))^2$$ which is minimized to $$(y_1-y_2+y_3)^2/3$$ by taking $$b_1=(y_1+2 y_2+y_3)/6$$ and $$b_2=(y_1 - y_3)/2$$. Because there is just one degree of freedom associated with the sum of squares, $$(y_1-y_2+y_3)^2/3$$ is also an estimate of $$\sigma^2$$.