- #1

Master1022

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- 117

- Homework Statement
- Given three random variables ## X##, ##Y##, and ## Z ## such that corr(X, Y) = corr(Y, Z) = corr(Z, X) = r, provide an upper and lower bound on ##r##.

- Relevant Equations
- Correlation

Hi,

I just found this problem and was wondering how I might go about approaching the solution.

Given three random variables ## X##, ##Y##, and ## Z ## such that ##\text{corr}(X, Y) = \text{corr}(Y, Z) = \text{corr}(Z, X) = r ##, provide an upper and lower bound on ##r##

I don't quite know how to attempt this rigorously, but I started by thinking about the problem geometrically. I don't know if this is incorrect, but I thought of the variables as 'vectors' and the correlations as the cosine of the angle between them.

Then, let us fix two vectors, ## X ## and ##Y##, with the angle between them ##\theta = \cos(r)##. Now we need to find the vector ##Z## such that it has the same angle between itself and both of ##X## and ##Y##. After doing some visualisation, I thought that the maximum angle possible between each 'vector' was 120 degrees (when the vectors are all planar), thus the correlation is ## cos(120^{o}) = -0.5 ##. Thus, my final answer would be: ## -0.5 \leq r \leq 1 ##: is that correct?

I suppose I could imagine this by imagining taking three pencils/pieces of spaghetti and standing them all up in a bunch. The angle between them all is 0 degrees, so correlation = 1. Then I could start increasing the angle between them all (as the pieces become less inclined to the table), so correlation is decreasing between them. The largest angle occurs (I think!) when they are all flat on the table, and the angle between them all must be equal to satisfy the constraints of the problem. Thus, that led me to the 120 degrees. Apologies for the silly explanation, but that is how I thought about the problem.

Thanks in advance

I just found this problem and was wondering how I might go about approaching the solution.

**Question:**Given three random variables ## X##, ##Y##, and ## Z ## such that ##\text{corr}(X, Y) = \text{corr}(Y, Z) = \text{corr}(Z, X) = r ##, provide an upper and lower bound on ##r##

**Attempt:**I don't quite know how to attempt this rigorously, but I started by thinking about the problem geometrically. I don't know if this is incorrect, but I thought of the variables as 'vectors' and the correlations as the cosine of the angle between them.

Then, let us fix two vectors, ## X ## and ##Y##, with the angle between them ##\theta = \cos(r)##. Now we need to find the vector ##Z## such that it has the same angle between itself and both of ##X## and ##Y##. After doing some visualisation, I thought that the maximum angle possible between each 'vector' was 120 degrees (when the vectors are all planar), thus the correlation is ## cos(120^{o}) = -0.5 ##. Thus, my final answer would be: ## -0.5 \leq r \leq 1 ##: is that correct?

I suppose I could imagine this by imagining taking three pencils/pieces of spaghetti and standing them all up in a bunch. The angle between them all is 0 degrees, so correlation = 1. Then I could start increasing the angle between them all (as the pieces become less inclined to the table), so correlation is decreasing between them. The largest angle occurs (I think!) when they are all flat on the table, and the angle between them all must be equal to satisfy the constraints of the problem. Thus, that led me to the 120 degrees. Apologies for the silly explanation, but that is how I thought about the problem.

Thanks in advance