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If I have a random vector in Rn that has a uniform distribution in the domain [a,b]n, a<0, b>0. Is uniformity lost or preserved (in the unit sphere) if I normalize the vector (using the euclidean norm)?

tinyhippo
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    Let $a=5,\ b=10$. Then, if $Y=\frac{X}{|X|}$, $P{Y_1<0}=0$ –  Apr 14 '15 at 08:45
  • I have altered the requirement on a and b – tinyhippo Apr 14 '15 at 09:07
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    Consider $n = 1, a = -1, b = 2.$ Denote the random variable uniformly distributed on $[-1,2]$ by $X.$ Since $P(X=0) = 0,$ we can define the random variable $Y = \frac{X}{|X|}.$ Now, observe that $P(Y = -1) = \frac13$ and $P(Y = 1) = \frac23.$ So normalization destroys uniformity. – jflipp Apr 14 '15 at 09:20
  • @G.Sassatelli's counterexample still holds, only slightly modified (pondering it would have been a more fruitful reaction). – Did Apr 17 '15 at 15:58

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