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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.

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    $\begingroup$ 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. $\endgroup$
    – rubikscube09
    Aug 25, 2021 at 14:41
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    $\begingroup$ You need to learn about copulas. $\endgroup$
    – g g
    Aug 25, 2021 at 15:43
  • $\begingroup$ @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? $\endgroup$
    – Kinka-Byo
    Aug 25, 2021 at 20:35

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