Consider a sample space consisting of colors of balls in an urn, denoted by $\Omega = \lbrace red, blue, green \rbrace$. Define a random vector $[X_{1}(w), X_{2}(w)]$, $w \in \Omega$ such that the random variables $X_{1}$ and $X_{2}$ have a correlation coefficient of 0.4. Note that $X_{1}, X_{2} \in \mathbb{R}$.
Using the brute force, it takes time to find $X_{1}$ and $X_{2}$ such that their correlation coefficient is 0.4. Is there an efficient way to do this?