I'm trying to figure out second moment of the following quantity


$$y = \frac{\langle x\_1, x\_2 \rangle}{\left\|x\_1\right\|\left\|x\_2\right\|}$$


Where $x\_1$, $x\_2$ are sampled independently from $\mathcal{N}(0, \Sigma)$


This can be solved exactly in 2-dimensions using algebraic manipulation: suppose eigenvalues of $\Sigma$ are $a$ and $b$, then


$$E[y^2] = \frac{a+b}{\left(\sqrt{a}+\sqrt{b}\right)^2}$$


Is there a similarly elegant expression for $n$ dimensions?


(update, I extended this formula to eigenvalues $a,b,c$ "by analogy" and it seems to hold numerically)



Second moment of cos(x,y) for Normal x,y?
This $y = \frac{<x\_1, x\_2>}{\|x\_1\|\|x\_2\|}$ is the distribution of the $cos \theta$ where $\theta$ is known as the canonical angle/principal angle of two random vectors $x\_1,x\_2$. $cos\theta$ is known as the canonical correlation between $X\_1,X\_2$ since we know their joint distribution $(X\_1,X\_2)$ from independence.


Usually we discussed when $(X\_1,X\_2)$ are assumed to be eigenvectors corresponding to a random matrix. And the derivation actually does not require normality assumption and can be extended to higher $r$, see [Anderson] Chap.10 for example. 


The derivation can be done as shown in [Anderson] using iterative optimization, however, there is a general treatment [Anderson2][Hsu] which derives the exact distribution of eigenvectors and therefore the moments associated under normality assumption of the sample.




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Update: I have answer a updated version of this question using directional statistics here, now it can be addressed by using projected normal distribution: <https://stats.stackexchange.com/questions/263896/moment-mgf-of-cosine-of-two-random-vectors>


[Anderson]An Introduction to Multivariate Statistical Analysis,1958,Wiley


[Anderson2]Multiple discoveries: Distribution of roots of determinantal
equations
T.W. Anderson


[Bartlett]Bartlett, M. S. "The general canonical correlation distribution." The Annals of Mathematical Statistics (1947): 1-17.<http://projecteuclid.org/download/pdf_1/euclid.aoms/1177730488>


[Hsu]Hsu, P. L. "On the distribution of roots of certain determinantal equations." Annals of Human Genetics 9.3 (1939): 250-258.


