# How does one generally find a joint distribution function (or density) from marginals when there is dependence?

So I know one can go from a joint density function $f(x,y)$ to marginal density functions, like $f_x(x)$ by integrating against the other variables as in $f_x(x) = \int f(x,y) dy$...but given $f_x(x)$ and $f_y(y)$ as densities for dependent random vars..how would one go about finding a joint density or distribution function?

Thanks

• In general, you can't recover the joint density from the marginals. You need more information. In a few cases (eg joint gaussian variables, or two bernoullis) the correlation would suffice. – leonbloy Oct 30 '11 at 22:27

For example, suppose the marginal densities for $X$ and $Y$ are both 1 on the interval $[0,1]$, 0 otherwise. One family of possibilities for the joint density is $f(x,y) = 1 + g(x) h(y)$ for $0 < x < 1$, $0 < y < 1$, 0 otherwise, for functions $g$ and $h$ such that $\int_0^1 g(x)\, dx = \int_0^1 h(y)\, dy = 0$, $-1 \le g(x) \le 1$ and $-1 \le h(y) \le 1$. And there are infinitely many other possibilities.
• ...Just like one obtains the joint distribution of every random couple (X,Y) from the marginal distribution of X and the conditional distribution of Y conditionally on X. I see no advantage here, rather a reformulation of some well-known characterization. Likewise, the WP page mentions Fréchet-Hoeffding copula bounds, which are (i) unsourced and (ii) a trivial translation of $P(\cap A_i)\le\min P(A_i)$ and $P(\cup A_i)\le\sum P(A_i)$. It seems that my previous question remains. – Did Dec 30 '11 at 18:29
• If you have a marginal distribution for $X$ and $Y$ and a conditional distribution on $Y$ given $X$, you might get two different marginal distributions on $Y$. Copulas give you function that maps a family of marginal distributions into a joint distribution. Conditional distributions don't d that. – Michael Greinecker Dec 30 '11 at 20:26