# Joint Probability Density of correlated uniform random variables

let X and Y be two continuous random variables with uniform distribution in the same interval [a;b].

They are correlated with a given correlation coefficient $$\rho$$.

I'm curious if there is the analytical opportunity to evaluate the joint probability density $$f(x,y)$$.

It is equal to $$f(x,y) = f(x)\cdot f(y|x)$$ but suppose $$f(y|x)$$ is unknown.

There are similar questions but:

• many of them (for instance 1, 2) refer to the standard distribution
• others tell that knowing $$\rho$$ is not sufficient to find the conditional probability density

Well, if it is true, I wanted to know which is the missing information and understand why $$\rho$$ is not sufficient. Maybe, there is the opportunity to estimate the missing part. Or maybe there exists some specific properties of the uniform distribution.

• It is not sufficient roughly speaking because correlation is only a measure of linear dependence. It is sufficient in the case that $X,Y$ are jointly normal. One can find examples of variables that have zero correlation but are not independent. Aug 25, 2021 at 14:41
• You need to learn about copulas.
– g g
Aug 25, 2021 at 15:43
• @g g I'm going to take a look. Just a question (since it seems a quite complex topic for a newbie like me): what can I get with copulas? Is it sufficient to get the conditional density? Does it allow me get the joint moments? Aug 25, 2021 at 20:35