This can be assumed as a follow-up question for this one.
Assume that $X$ is a truncated Gaussian random variable on range $[a,b]$, $E[X]=\mu$ and $Var(X)=\sigma^2$. Find a function of $X$ that converts it to a uniform random variable on the range $[l,u]$.
I have been working on the problem but could not find the solution. I believe we can use the same techniques as those used for this one. So, we should use the CDF of the truncated normal variable which is given here. I don't know how to use it though.
Let $F$ denote the CDF of $X$. Then the random variable $Y = F(X)$ is standard uniform. After all, \begin{equation} \begin{split} P(Y \leq y) &= P(F(X) \leq y) \\ &= P(X \leq F^{-1}(y)) \\ &= F(F^{-1}(y)) \\ &= y \end{split} \end{equation} for all $y \in [0, 1]$. So your desired function is just the CDF of the truncated normal distribution. Unfortunately, this cannot be written in closed form.