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Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$.

And Y is a random variable with Beta distribution over domain $[0,1]$, with PDF function $f_Y$.

Seeking a map $T: \mathbb{R} \rightarrow [0,1]$ such that $f_X(x) = f_Y(T(x))$.

In words, if X is "re-distributed" over [0,1] then it would be a Beta distribution.

In other words, $T$ squeezes the domain from $(-\infty, +\infty)$ to [0,1].

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