Given x ~ N(0,D) is an n-dimensional normal random vector, where D is a diagonal matrix with positive entries. Prove that
$E [ x x^T / (x^Tx)] = D^(1/2) / trace(D^(1/2))$,
Or equivalently, let y = x/||x||, what is the covariance of y ?
Hello, I'll empirically answer your first question via an example in [R]. After, you can try to apply it to any specific problem.
...
Get a vector v={1,2,...,5} and a second one v2={6,...,10}.
Using R, you can use the function normalize.vector(), from the library "ppls"
You shall get two normalized vectors (nv1,nv2).
Then use the function "cov()", passing three paramethers: "nv1" , "nv2" and a method ("kendall" or "spearman").
To test it, just use "cov()" with the original vectors.
For more details you can type ?cov and read the documentation. If you wanna go deeply, try the reference: "Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole."
Hope It was useful.
Hi @Joaquim, thanks, but I have already run simulations to empirically verify the claim. Now I need to prove it.
Ohh, ok. In this case I think you should change the title to "How to prove..." or "How to prove this...". This will increase the odds of getting a more accurate answer.
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