CDF on Standard uniform gives the same distribution

Question

Assume that $X$ has a continuous and strictly increasing CDF $F_X$. Define $Y = F_X^{-1}(U)$ where $U$ is standard Uniform. How dow I show that $X$ and $Y$ have the same distribution?

Answer 1

We have $$F_Y(y)=\Pr(Y\le y)=\Pr(F_X^{-1}(U)\le y).$$

Because $F_X$ is strictly increasing, $$\Pr(F_X^{-1}(U)\le y)=\Pr(U \le F_X(y))=F_X(y).$$