Construct random variables with given correlation

Question

I looked at an old exam and there was a question to construct two random variables $X$ and $Y$, such tha

$$\rho_{X,Y} = \frac{1}{3}$$

and I have no clue how to attempt this problem. I've seen this post, but was thinking if there was an easier way if the random variables doesn't have to be normal-distributed.

Answer 1

Take i.i.d. random variables $U,V$ with standard normal distribution and take $X=U, Y=aU+bV$. can you fins $a,b$ such that $\rho_{X,Y} =\frac 1 3$?

Answer 2

Guide:

Let $U,V,Y$ be iid random variables with variance $1$.

Then for $X=U+V+Y$ you can find that $\mathsf{Cov}(X,Y)=1$ by using bilinearity of covariance.

Also find $\mathsf{Var}(X)$ and draw conclusions.