reverse engineering with correlation between two random Variables

Question

If some one give me correlation between two random variables $X$ and $Y$, then how I can find the distribution of $X$ and $Y$.?

Answer 1

very easy you can't. For example if I know that $Cor(X,Y)=0$ then there are lots of possible choices for the distribution of $X$, the distribution of $Y$ and also for the joint distribution of $(X,Y)$. To give you some examples:

• $(X,Y)$ could have a two dimensional normal distribution with uncorrelated components
• $X$ could have a uniform distribution on $(-1,1)$ and $Y=X^2$. In this case $X$ and $Y$ are uncorrelated but not independent.
• $X$ and $Y$ could have any distribution (normal, exponential, binomial, poisson,...) and be independent. In this case their correlation is also zero.

In general you need to know the marginal distributions of $(X,Y)$ (i.e the distribution of $X$ and the distribution of $Y$) and in addition you need to know the dependence structure (aka Copula) of $X$ and $Y$ to be able to get the joint distribution (see Sklar's Theorem).

The correlation does not even specify the dependence structure (only in special cases, when e.g. the Copula is a Gauß Copula).