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How could I convert a non-uniform random variable distribution to a uniform distribution?
Is there a formula for such a transformation?

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    The question to ask: what distribution does your "non-uniform random variable" follow? Then you can just apply the corresponding CDF to the nonuniforms.2010-08-28
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    I think [copula](https://en.wikipedia.org/wiki/Copula_(probability_theory)) are a related concept to what you want (generalized to multivariable distributions).2015-08-22

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Use the probability integral transform.