X=F^{-1}(U),

$$

then $X$ ia a random variable with CDF $F_X(x)=F(x)=\textrm{Prob}\{X\le x\}$.

then $X$ is a random variable with CDF $F_X(x)=F(x)=\textrm{Prob}\{X\le x\}$.

We'll verify this in the special case in which $F$ is continuous and bijective so that its inverse function exists and

can be denoted by $F^{-1}$.

We'll verify this in the special case in which $F$ is continuous and bijective so that its inverse function exists and can be denoted by $F^{-1}$.

Note that