# 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. – StubbornAtom 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
• Please explore stats.stackexchange.com/search?q=probability+integral+transform* for answers. – whuber Nov 25 '18 at 16:07
• Google 'probability integral transform'. – StubbornAtom 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 – Glen_b -Reinstate Monica Nov 26 '18 at 2:04

By the probability integral transform, the CDF of $$X$$ has a uniform distribution. Thus,
$$Y = u(X) = \frac{1}{4}(X - 1)^2$$