Prove that $\forall x[-1, 1]$ exist two random variables such that their correlation is $x$

Question

The task is to prove that $\forall x\in[-1, 1]\ \ \exists \xi, \eta$ - random variables such that $\text{Corr}(\xi, \eta)=x$.

I've tried using the following formula: $$\text{Corr}(\xi, \eta)=\frac{\text{Cov}(\xi, \eta)}{\sqrt{\mathbb D\xi}\sqrt{\mathbb D\eta}}=\frac{\mathbb E(\xi \cdot \eta) - \mathbb E\xi \mathbb E\eta}{\sqrt{\mathbb D\xi}\sqrt{\mathbb D\eta}}$$ While $\mathbb E \xi, \mathbb E\eta, \mathbb D\xi, \mathbb D\eta$ can take any value $\in \mathbb R$ independently of each other, I'm not sure how to deal with $\mathbb E(\xi \cdot \eta)$.

Answer 1

If $X$ and $Y$ are i.i.d. with a standard normal distribution and $\xi=aX+bY, \eta=Y$ then $corr(\xi,\eta)=\frac b {\sqrt {a^{2}+b^{2}}}$. For any $x \in [-1,1]$ we can choose real numbers $a,b$ such that $\frac b {\sqrt {a^{2}+b^{2}}}=x$. [Take $b=x,a =\sqrt {1-x^{2}}$].