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Let

$Y_1=\beta_1+\epsilon_2$

$Y_2=2\beta_1-\beta_2+\epsilon_2$

$Y_3=\beta_1+2\beta_2+\epsilon_3$

where $\epsilon_1$,$\epsilon_2$, and $\epsilon_3$ are uncorrelated random variables with $E(\epsilon_i)=0$ and $Var(\epsilon_i)=\sigma^2$ for $i=1,2,3$

I need help finding $Var(\hat{\beta}_1)$ and $Var(\hat{\beta}_2)$

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Assuming $\hat \beta_1$ and $\hat \beta_2$ means least square estimates and using there values from your earlier question Find the Least Squares Estimates, we have$$\hat \beta_1=\frac{y_1+2y_2+y_3}{6}, \hat \beta_2= \frac{-y_2+2y_3}{5}$$
Now as $\epsilon_i $ are uncorrelated we have $y_i$'s as uncorrelated and then by using the fact that for uncorrelated variables $X_i$'s, $Var(\sum X_i)=\sum Var(X_i)$ and that $Var(aX)=a^2Var(X)$, we have $$Var (\hat \beta_1)=\frac{1}{36}(Var(y_1)+4Var(y_2)+Var(y_3))=\sigma^2/6$$ Using $Var(a+X)=Var(X)$, in the last equality. Similarly $Var(\hat \beta_2)=\sigma^2/5$.

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