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I can think of the "Universality of the Uniform":

Given an Unif(0,1) r.v., we can construct an r.v. with any cts distribution we want. Conversely, given an r.v. with an arbitrary cts distribution, we can create a Unif(0,1). ----From Introduction to Probability by J.K.Blitzstein & J.Hwang.

Other than this "Universality of the Uniform" or "Fundamental theorem of Simulation", is there any other (potentially in more depth) explanation?

Thanks in advance.

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    $\begingroup$ have you seen the inverse cdf sampling method? $\endgroup$ – Chinny84 Jun 8 '15 at 13:48
  • $\begingroup$ @Chinny84, hi thanks for comment. Yes, in some way, basically that is where we use uniformly r.v. to construct other r,v.? $\endgroup$ – Bob Jun 9 '15 at 5:21

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