Problem: Random variable $X$ has uniform distribution over $[0,1]$ and $Y=\min(X,1-X)$. Prove that the expected value of $Y$ is $\frac 14$.
Attempted Solution: I start by finding the CDF:
$F_Y(x) = P(Y<x) = 1 - P(\min(X,1-X)>x) = 1 - P(X>x)P(1-X>x)$
Then by finding $P(X>x)$ and $P(1-X>x)$ , I could calculate the derivative of $F_y(x)$ and thus find the PDF of $Y$. Is this the right approach?
Also is $P(X<x)$ equal to $1-x$?