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If $X$ is a continuously distributed random variable and $F$ is the c.d.f. of its distribution, then $F(X)$ is uniformly distributed in the interval $(0,1).$

While i'm clear with the mathematical proof, I'm looking for an intuitive answer to this.

Michael Hardy
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LUCIFER
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    This must be a duplicate, but cannot find it – kjetil b halvorsen Aug 21 '17 at 22:29
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    Please give a tiny bit of hint about what you don't find intuitive. Are you comfortable with CDF's in general but uniform RVs somehow baffle you, or the opposite, or both? – kimchi lover Aug 21 '17 at 22:41
  • This only applies to continuous cumulative distribution functions, i.e. for continuous random variables – Henry Aug 22 '17 at 00:14
  • What is the probability for an outcome to occur in an event whose probability is $y$? – Graham Kemp Aug 22 '17 at 00:18
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    It's really unclear what this is about. To ask for an intuitive explanation, it helps to give a clear statement of the mathematical fact you are interested in--the definition, formula, whatever, saying what all the parts of it are; explain the way in which you already understand it (in this case, what is the mathematical proof?), and then ask for a more intuitive explanation. If you want to try this, use the "edit" link just below the question text to insert this information into the question; don't try to explain in comments. – David K Aug 22 '17 at 00:31
  • @kjetilbhalvorsen - try https://math.stackexchange.com/questions/868400/showing-that-y-has-a-uniform-distribution-if-y-fx-where-f-is-the-cdf-of-contin or some of those linked to it – Henry Aug 22 '17 at 00:35
  • @DavidK : The language in which the posting was initially expressed was not up to passable standards of precision for mathematicians or even for homework being graded. But I have edited to clarify what I think must have been the poster's intent. – Michael Hardy Aug 22 '17 at 02:25
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    @MichaelHardy Yes, the language was so poorly organized that I completely missed the point. Now that you've helped OP out by fixing it, I think this is distinct from the linked question because this one asks for intuition rather than a proof. I'm voting to reopen. – David K Aug 22 '17 at 05:07

1 Answers1

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Assuming we are dealing with continuous random variables

  • the CDF $F(X)$ is a continuous increasing function of $X$ where the probability that $F(X) \le p$ for $p \in (0,1)$ is the probability that $X \le q$ when $F(q)=p$, which is $p$

  • so $F(X)$ has a uniform distribution on the unit interval, since the probability it is less than or equal to a given $p \in (0,1)$ is $p$

Henry
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