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The eigenvector matrix of a Wishart matrix is Haar distributed and that implies that the eigenvectors are uniformly distributed on a sphere. I'm interested to know what is the distribution of individual entries of this haar distributed eigenvector matrix?

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So you want the marginal distribution of one of the coordinates of a uniformly distributed point on the unit sphere in $\mathbb R^n?$ Isn't its square Beta distributed, with parameters $(1/2, (n-1)/2)$?

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