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My Task is the following! $X_1 \sim Ex(2)$ and $X_2 \sim Ex(1/2)$ find a distribution so that $\rho_{\tau}=-0.85$. I have a little problem finding this distribution. Has anyone a clue what copula I can use?

$\rho_{\tau}=4*\int_0^1\,C(u1,u2)\,dC(u1,u2)-1$

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The marginal distributions are independent of the step of choosing the copula to satisfy the condition, so ignore the margins for now. To begin with this is purely a matter of the relationship between $\rho_\tau$ and the copulas you will already know about.

There will be an infinite number of potential copulas you could choose (if you've seen even a few common choices you'll doubtless already know some that are suitable); start by choosing one with a single parameter, where the Kendall correlation coefficient is related to that parameter in a simple way.

You'll have been given (or failing that, can easily find/derive*) formulas for $\rho_\tau$ for some standard copulas.

Several standard books (e.g. Nelsen's book) and review articles shortcut the process by simply listing the Kendall and Spearman correlations for many common copulas (e.g. FGM has $\rho_\tau=2\theta/9$, Clayton has $\rho_\tau=\theta/(\theta+2)$, Gaussian has $\rho_\tau=\frac{2}{\pi} \arcsin(\rho)$, etc), but I would bet that you will already have been given what you need in lectures/notes/examples, or will have derived a suitable one for an exercise already.

Note that the fact that the correlation is large negative will restrict the available choices somewhat (some copulas don't have $\rho_\tau$ go down to that); you need to pay attention to both the permissible values of the parameter of the copula as well as its relationship to $\rho_\tau$.

* e.g. via the formula you gave

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