A random variable $X$ has the PDF

$f_X(x) = \frac{x - 1}{2}, \ 1 < x < 3$

Find a monotone function $u(x)$ such that the variable $Y = u(X)$ has the distribution $Uniform(0,1)$.

  • $\begingroup$ Any thoughts to share on the problem? If this is homework, consider adding the self-study tag and read the tag wiki. $\endgroup$ Nov 25 '18 at 15:51
  • $\begingroup$ I don't have any valuable ideas. I tried to reverse the multiplication and addition, but I cannot find a correct transformation. $\endgroup$
    – MSE
    Nov 25 '18 at 15:58
  • 1
    $\begingroup$ Please explore stats.stackexchange.com/search?q=probability+integral+transform* for answers. $\endgroup$
    – whuber
    Nov 25 '18 at 16:07
  • $\begingroup$ Google 'probability integral transform'. $\endgroup$ Nov 25 '18 at 16:08
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    $\begingroup$ MSE you can answer your own question (I have reopened so that you can choose to do so), though it's possible that it's already answered on site $\endgroup$
    – Glen_b
    Nov 26 '18 at 2:04

By the probability integral transform, the CDF of $X$ has a uniform distribution. Thus,

$Y = u(X) = \frac{1}{4}(X - 1)^2$


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