# Transform X to get Y such that Y has a Uniform(0,1) distribution

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 $$Uniform(0,1)$$.

• Any thoughts to share on the problem? If this is homework, consider adding the self-study tag and read the tag wiki. Nov 25 '18 at 15:51
• I don't have any valuable ideas. I tried to reverse the multiplication and addition, but I cannot find a correct transformation.
– MSE
Nov 25 '18 at 15:58
– whuber
Nov 25 '18 at 16:07
• Google 'probability integral transform'. Nov 25 '18 at 16:08
• 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 Nov 26 '18 at 2:04

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

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