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I am working on a problem: $Y = A+B+C$. I know that $Cov(A,B+C)=0$,$Cov(B,C)=0$, $E(B)=0$ and $E(C)=0$. Can I get the $Cov(A,B)=0$? Now I can only get $Cov(A,B)=-Cov(A,C)$. But it seems to be intuitive that both $Cov(A,B)$ and $Cov(A,C)$ should be zero?

Xuanqi Liu
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For a counterexample, suppose $var(B) = var(C) > 0$, and $A = B - C$.

Then $$ \begin{aligned} Cov(A, B+C) &= Cov(B-C, B+C) \\ &= E((B-C)(B+C)) \\ &= E(B^2 - C^2) \\ &= E(B^2) - E(C^2) \\ &= var(B) - var(C) \\ &= 0 \end{aligned}$$ but $$\begin{aligned} Cov(A,B) &= Cov(B-C, B) \\ &= E((B-C)B) \\ &= E(B^2 - BC) \\ &= E(B^2) - E(BC) \\ &= var(B) - Cov(B,C) \\ &= var(B) \\ &> 0 \end{aligned}$$