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The following is from an optional exercise guide. The fact that the result should be a uniform distribution was given to me as a hint by a hasty professor, nevertheless I have been unable to solve it. The problem is:

Given $Z = (Z_1,...Z_d)$ $\sim$ $N(0,\mathbb{I}_d)$, take $X := \frac{1}{||Z||}(Z_1,...Z_d)$.Find the distribution of X

My initial idea was to attempt a change of variables, however the function $g:\mathbb{R}^d \rightarrow \mathbb{R}^d$ such that g(Z) = X is only injective on $S^{d-1}$

Lucas G
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  • I think it is straightforward to show that $X$ is rotation-invariant (due to $Z$ being rotation-invariant). But I am not sure how easy it is to show that the only rotation-invariant distribution on $S^{d-1}$ is the uniform distribution; some discussion here. – angryavian Mar 21 '24 at 03:12

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