# How to construct a joint pdf, other than the independent coupling, given two marginal pdf's?

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 $$f_{U,V}$$.

• Have you tried anything? Commented Jun 4, 2020 at 19:35
• 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. I really don't see how to approach this question. Commented Jun 4, 2020 at 19:43

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.
• Thank you for your answer. So far as I understand it, $F_U (U), F_V(V) \sim U(0,1)$ and the joint distribution of $(F_U(U), F_V(V))$ is a copula. But I don't understand your last paragraph. I think you are saying the following. Let $(X,Y)$ be distributed according to a copula $C$. Then $F^{-1} _U (X) \sim F_U$ and $F^{-1} _V (Y) \sim F_V$ where $F_U, F_V$ are the desired cdf's. Then $(F^{-1} _U (X), F^{-1} _V (Y) )$ follows a joint distribution that can be found through the copula $C$. Is my understanding correct? Commented Jun 4, 2020 at 21:00