Uniqueness of Bivariate Probability Distribution I am studying some intermediate level probability theories. And I am wondering what can characterize an unique bivariate probability distribution. I mean, what kind of conditions given can determine a unique bivariate probability distribution. For example, I know the margins of the distribution as well as the distribution of the maximum, can I deduce a unique bivariate probability?
Let's say, I want to determine the bivariate distribution function of $\textbf{X}=(X_1,X_2)$ and I know, for example, $X_i$ is distributed Pareto and the maximum of $X_i$ is also distributed Pareto, with these information, can I deduce a unique joint distribution function?
To make a concrete example. For example, if I know $X_i$ is distributed $F(x_i)=1-T_i x_i^{-\theta}$ and the maximum is distributed $F(x)=1-\tilde{T}x^{-\theta},$ where $\tilde{T}=(\sum_i T_i^{1/(1-a)})^{1-a}$, can I prove that $1-(\sum_i (T_ix_i^{-\theta})^{1/(1-a)})^{1-a}$ is the only joint distribution of $X_1$ and $X_2$?
More generally, if we know $\Pr(X_1\leq x_1), \Pr(X_2\leq x_2)$ and $\Pr(X_1\leq x, X_2\leq x)$, can we know $\Pr(X_1\leq a, X_2\leq b)$ for any a, b?
 A: One situation in which $X$ and $Y$ both have Pareto distributions, is when $\Pr(X=Y)=1,$ and another is when $X$ and $Y$ are independent.  Those are two very different joint distributions of the pair $(X,Y)$, so that bivariate distribution is not characterized by its two margins.
I don't off hand know an example in which $(X_1,Y_1)$ and $(X_2,Y_2)$ both have the same margins and also the same distribution of the maximum of the two but have different joint distributions, but I will be surprised if it turns out not to be easy to construct such examples.  Maybe I'll add one here later.
One special case that often arises is the bivariate normal distribution.  There are cases in which $(X,Y)$ has standard normal margins and $X,Y$ are uncorrelate but $(X,Y)$ does not have a bivariate normal distribution.  However, when the pair does have a bivariate normal distribution, then the margins and the correlation characterize that distribution completely.
There are very numerous joint distributions in which the margins are uniformly distributed on $(0,1)$.  Those are called copulas, and you can google that term.
