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Can one consider that a general probability distribution for some random variable X over a Euclidean space is actually the probability distribution of a uniform random variable over a curved 1D-space ?

(I was thinking about a map of the type $\frac{1}{\sqrt{1+f'^2}}$ between the two distributions)

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    You might want to explain your idea of a curved 1D-space. Extrinsic curvature of a curve is allright (given an embedding) but you would need intrinsic curvature, which does not exist... Or am I mistaken?2011-07-12
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    @Didier Piau : Suppose you have a function y=f(x) which is not necessarily invertible. Suppose now that s is the curvilinear abscissa along this curve (The map I was talking about then gives the metric of the 1D-space) If X is a random variable with uniform probability distribution over s, does it not induce an apparent, non-uniform, probability distribution over x ? Sorry if this sounds fuzzy, I'm not very good at formalizing intuitions...2011-07-13

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