I don't understand how the P[U<=F(x)] = F(x)can someone explain this?
//Comment
This can be said because it was given to us that the function is a "non-decreasing function" and that U is a uniform random variable. This is enough information to make this step. So,
- $ \begin{align} & {} \Pr(U \leq F(x)) = F(x)\quad \text{(because }\Pr(U \leq y) = y,\text{ since }U\text{ is uniform on the unit interval)} \end{align} $
