Mapping Normal distribution to a new bounded distribution. My question is vague. I am looking for distributions that can be obtained from Normal distribution by mapping it to a bounded interval.
"By mapping" I mean that a probability distribution function(PDF) is obtained from PDF for Normal distribution by applying some function. For example, log normal distribution is obtained from Normal distribution by using exponential function.
What distribution do you know that obtained from Normal distribution by mapping to a bounded interval?
 A: I'm not sure if the question makes much sense because of this:
Let $X$ be any continuous random variable (bounded or not) and $F_X$ its CDF. Then the random variable $U=F_X(X)$ has uniform distribution on $[0,1]$. 
Now pick any other continuous random variable (bounded or not), say $Y$ and let $F_Y$ be the corresponding CDF. Then the random variable $F_Y^{-1}(U)$ has the same distribution as $Y$ has. 
Concluding, if you pick $X$ to be normal, you can generate any arbitrary continuous distribution $F_Y$ by considering the random variable, $$F_Y^{-1}\left(F_X(X)\right).$$
A: Every distribution on $\mathbb R$ (or on any interval of $\mathbb R$) is the distribution of a random variable $H(X)$, where the random variable $X$ is standard normal, for some nondecreasing function $H$.
Proof: Let $F$ denote the CDF of the desired distribution and $G$ the standard gaussian CDF. Then $G(X)$ is uniform on $(0,1)$ and, for every $U$ uniform on $(0,1)$, $F^{-1}(U)$ has CDF $F$. The inverse $F^{-1}$ must be defined with care when the CDF $F$ of the target distribution is not increasing but only nondecreasing, but this can be done, and it works. By composition $H=F^{-1}\circ G$ fits.
