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Suppose the correlation between vectors $A$ and $C$ of size $N$ is some value $r_{AC}$, and likewise for $B$, $C$, and $r_{BC}$. What is the distribution for $r_{AB}$?

Because correlations are cosines of angles between unit vectors, it is easy enough to compute the minimum and maximum potential values for $r_{AB}$. I have an intuition that this can be extended by calculating (analytically) the distribution of angles from some vector to some hypersphere, but I don't know how to start.

Mark Miller
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  • Have you seen this question and its answers? – Dilip Sarwate Feb 23 '23 at 01:24
  • I had not, but they only appear to indicate the bounds, not the distribution, as far as I understand the answers. – Mark Miller Feb 23 '23 at 05:38
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    Unless you specify some distributions for $A,B,C$, your question regarding the distribution is meaningless; $r_{AB}$ is a constant that satisfies bounds given in the answers to the question cited. Also, read the answer by whuber more carefully, especially the section titled "Proof of Sufficiency of Conditions" where he assumes that the vector components are iid standard normal random variables. – Dilip Sarwate Feb 23 '23 at 14:14

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