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Suppose z is a continuous random variable and follows a certain distribution

We define φ(x)=p(z>=x), where x is a continuous random variable

It can be proved that the continuous random variable φ(x) follows the uniform distribution: p(φ(x)<=φ(ξ)) =p(x>=ξ) =φ(x)

However, the distribution of φ(x) should depend on the distribution of x but the proof procedure has nothing to do with it. That's what I am confused about.

To elaborate on my point, I am confused about the process of getting different values of φ(x) and then the distribution of it. The value of φ(x) really depends on the value of x, which is not specified in the proof.

Thanks in advance!!

Yuan
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  • Careful: "it can be proved" is incorrect unless you assume $Z$ follows a *continuous* distribution. – whuber Jan 05 '20 at 22:06
  • @jbowman Thank you, I read the post carefully. I think I understand how to extend the formula to the null hypothesis. What I am confused about is really the process of getting different value of 1-CDF. – Yuan Jan 05 '20 at 22:46
  • @whuber, thank you! edited the question – Yuan Jan 05 '20 at 22:46
  • I don't understand the question, because "$x$" refers to a *number* but you write as if it had a "distribution." Note, too, that $\phi(X)$ is not a continuous random variable unless $X$ is. You also refer several times to "the proof" but you haven't exhibited anything that looks like a proof. – whuber Jan 05 '20 at 23:30
  • @whuber The proof I referred to is: p(φ(x)<=φ(ξ)) =p(x>=ξ) =φ(x). x should be a random variable. For example, in the regression setting, x should be the test-statistics through repeated sampling. – Yuan Jan 06 '20 at 00:02
  • In that case your definition of $\varphi$ is incorrect. You appear to be trying to define the survival function of the random variable $Z$ and in that definition $x$ *must* be a number. The reason I see no kind of proof here is that you provide only a sequence of equalities but (a) not all terms are defined (what is "$\xi,$' for instance?) and (b) no reasoning is provided. Both (a) and (b) are essential parts of proofs. – whuber Jan 06 '20 at 00:07
  • Thank you, do you mean x should be a fixed number? But x is the independent variable in the function. May you explain more? thank you so much! – Yuan Jan 06 '20 at 00:22
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/102921/discussion-between-yuan-and-whuber). – Yuan Jan 06 '20 at 00:33

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