If $X$ is uniformly distributed over $(a,b)$ find a random variable which is uniformly distributed over $(0,1).$

Question

If $X$ is uniformly distributed over $(a,b)$ find a random variable which is some function of $X$ that is distributed as $U(0,1)$.

How can I find this??

Answer 1

Let's do it in two steps. Firstly, the random variable $$Y:=X-a$$ is uniformly distributed on $(0,b-a)$, i.e. $Y\sim U(0,b-a)$. Secondly, to get the upper bound equal to $1$ we need to divide $Y$ with $b-a$ (these are all calculations with constants, so they live the distribution unchanged). Hence, let $$U:=\frac{Y}{b-a}$$ then $U\sim U(0,1)$. To get $U$ in terms of $X$ write $$U=\frac1{b-a}Y=\frac{X-a}{b-a}$$