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Corr(A,B)=0 -> Cov(A,B)=0 Corr(B,C)=0 -> Cov(B,C)=0
Cov(A,C)=E(AC)-E(A)E(C) Cov(A,B)=E(AB)-E(A)E(B)=0 -> E(AB)=E(A)E(B) Cov(B,C)=E(BC)-E(B)E(C)=0 -> E(BC)=E(B)E(C)
To prove that Corr(A,C)=0 the variables should be independent. While independence implies uncorrelatedness, uncorrelatedness does not imply independence.
If the variables are independent, then it naturally follows that Corr(A,C)=0
So assuming the variables are independent, can you still prove it mathematically?
I understand the formulas and logic you provide, but isnt there a way to prove it using the formulas?
If X and Y are independent, then the following holds for expectations
E(XY) = E(X)E(Y)
Hence, it follows directly from the covariance formula that Cov(A,C)=0.
Alternatively, Cov(A,B)=E(AB)-E(A)E(B)=0 -> E(AB)=E(A)E(B) -> E(A) = E(AB)/E(B) Cov(B,C)=E(BC)-E(B)E(C)=0 -> E(BC)=E(B)E(C) -> E(C) = E(BC)/E(B)
Cov(A,C)=E(AC)-E(A)E(C)=E(AC) - [E(AB)/E(B)]*[E(BC)/E(B)]=E(AC)-E(AB)E(BC)/(E(B))^2
Given that that the variables are independent, the expression reduces to Cov(A,C)=E(AC)-E(AC)=0
Since Corr(A,C) = Cov(A,C)/st.dev(A)*st.dev(C), it follows that Corr(A,C)=0
The alternative route is redundant, though, once you know that the variables are independent.
Hope it helps.
...how are you coming to the conclusion that A is independent of C? The fact that cov(A,B) = 0 and cov(B,C)=0 only tells you that A is independent of B and C is independent of B but you don't necessarily know the relationship between A and C. The relationship could actually be A=C. You're assuming that step it seems, and as a result you can't really say whether cov(AC) would or wouldn't be 0
Come on, NY, read the math! You can derive the result from Cov(A,B)=0 and Cov(B,C)=0 (see the alternative route). However, you need to make the assumption of independence.
The fact that Cov(A,B)=0 does not mean that A is independent of B! However, if two variables are independent, it implies zero covariance.
corr(a,b)=corr(b,c)=0 Set correlation equations equal to each other (A-E(A))(B-E(B))/(s.d(A))(s.d(B))=((B-E(B))(C-E(C))/(s.d(b))(s.d(c))) Multiply both sides by s.d(B)/(B-E(B)) and get: (A-E(A))/s.d(A)=(C-E(C))/s.d(C) Multiply both sides by (A-E(A))/s.d(A)) and you get (A-E(A))^2 / (s.d. (A))^2 =((A-E(A))(C-E(C))/(s.d(A))(s.d(C))) Which is equal to Var(A)/Var(A) = corr(A,C) Therefore corr(A,C)=1
Someone please tell me if you see a flaw, but it looks good to me
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