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When generating uniformly-distributed samples from a multidimensional distribution, I believe that sampling each dimension independently produces uniformly-distributed samples from the original distribution.

For example, If I want to generate uniformly-distributed samples of 3D points, I can sample each axis from the uniform distribution and then combine the results. Sampling dimensions independently from uniform distributions produces results that are also sampled from a uniform distribution.

Is there a name for this principle?

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    $\begingroup$ I'd say this is an application of Fubini - or just the definition of product measure $\endgroup$ – Hagen von Eitzen Feb 20 at 3:01

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