Let $X$ and $Y$ be continuous random variables with probability density function as $p_x(X)$ and $p_y(Y)$. If $X$ and $Y$ are related by an invertible function $f$ as $f(X)=Y$, then using change of variable trick, we know that
$p_x(X) = p_y(Y) \left| det( \frac{\partial Y}{\partial X} ) \right| $
where det(.) is a determinant of a matrix.
Question: If $p_x(X)$ and $p_y(Y)$ are fixed, then is $f$ unique?
For example, suppose $X$ is Gaussian random variable with $\mu=0, \sigma=1$ and $Y$ is beta random variable with $\alpha=\beta=2$, then can we say that there exist a unique invertible function $f$ which converts $X$ to $Y$?
