I am trying to find joint densities for which the marginal densities are $f_U(u) = 2\exp(-2u), u\geq 0$ and $f_V (v) = \exp (-v), v \geq 0 $. Of course I can take the joint pdf $f(u,v) = f_U(u) \cdot f_V(v)$ (by assuming independence). But how can I construct other joint densities for which the marginals are $f_U$, $f_V$?
Attempts so far
Attempt 1 I have tried to construct a joint pdf along a line. Something like $f(x,y) = \exp (-y) 1 _{ \{ 2y = x\} } (x,y)$. But that does not seem to do the trick because $f$ integrates to 0.
Attempt 2 Let $X_1, X_2 \sim \text{EXP} (1)$ be independent random variables and put $U=X_1$ and $V=\text{min} \{ X_1, X_2 \}$. Then the joint pdf will be
\begin{align*}
f _{U,V} (u,v) = \frac{\partial ^2}{\partial u \partial v} \mathbb{P} (U\leq u, V\leq v) = \frac{\partial ^2}{\partial u \partial v} \mathbb{P} (X_1\leq u, \min \{ X_1, X_2 \} \leq v).
\end{align*}
I do not know how to proceed further. I think I have to integrate $f_U (x) f_V (y)$ over some area $A (u,v) \subseteq \mathbb{R} ^2$ in order to obtain $\mathbb{P} (U\leq u, V\leq v)$ and then take the double partial derivative to find $f_{U,V}$.
Best Answer
Let $F_U(u), F_V(v)$ be the c.d.f.s corresponding to the given marginal p.d.f.s $f_U(u), f_V(v).$ Then $F_U(U)$ and $F_V(V)$ (note that the arguments are denoted by capital letters this time, so these are random variables) are uniformly distributed in $[0,1].$ Those are their marginal distributions. Their joint distribution is an example of a copula. A copula is any probability distribution of a random variable taking values in the cube $[0,1]^n$ whose marginal distributions are uniform in the interval.
Given any copula, you can take any of the marginals – call it $X$ – and let $U= F_U^{-1}(X),$ where $F$ is a continuous c.d.f., and you $F_U$ is then the c.d.f. of the random variable $U$ that is defined that way.
So the question is: what copulas exist? See the linked Wikipedia article.