What exactly determines a bivariate distribution in statistics?

[covariance]

I'm having some trouble wrapping my head around multi-variate distributions. Most textbooks describe it by starting off with two random variables $$X, Y$$ and introducing $$P(X \leq x, Y \leq y)$$. This initially led me to believe that $$X, Y$$ uniquely determine the distribution over $$R^2$$ - I later confirmed with my TA that this isn't true and that one must explicitly specify the cdf on $$R^2$$. Moreover one cannot determine mutual independence given only the distributions, and there exist both dependent and independent distributions for any given pair of random variables.
Since this is the case, wouldn't it be better to describe $$(X, Y)$$ as a map from the sample space to $$R^2$$ and require that the marginal distributions be equal to X and Y? Or have I got the concept somewhat wrong? Some books also casually define random variables as functions of multiple random variables (e.g. $$Z = g(X, Y)$$) and this is a bit confusing too. Don't I need the distribution on $$(X, Y)$$ to get anything useful out of $$Z$$?

 

[Valkarie]

Any clarification would be great. Thanks!A:<blockquote><p>Most textbooks describe it by starting off with two random variables X, Y and introducing P(X ≤ x, Y ≤ y). This initially led me to believe that X, Y uniquely determine the distribution over R2 - I later confirmed with my TA that this isn't true and that one must explicitly specify the cdf on R2.</p></blockquote>Yes, this is correct. The joint distribution of $X$ and $Y$ does not determine the marginal distributions of $X$ or $Y$, so you have to specify all four of the marginal and joint distributions when describing the multivariate distribution.<blockquote><p>Moreover one cannot determine mutual independence given only the distributions, and there exist both dependent and independent distributions for any given pair of random variables.</p></blockquote>Yes, this is correct. The joint distribution tells you both the marginal distributions as well as any dependence between them.<blockquote><p>Since this is the case, wouldn't it be better to describe (X, Y) as a map from the sample space to R2 and require that the marginal distributions be equal to X and Y?</p></blockquote>No, this is not the same thing. A map from the sample space to $\mathbb{R}^2$ implies that the joint distribution of $X$ and $Y$ is completely determined. Furthermore, it implies that $X$ and $Y$ are independent, since knowing the value of either does not tell you anything about the value of the other. However, we typically don't assume that $X$ and $Y$ are independent, so this description is too restrictive.<blockquote><p>Some books also casually define random variables as functions of multiple random variables (e.g. Z = g(X, Y)) and this is a bit confusing too. Don't I need the distribution on (X, Y) to get anything useful out of Z?</p></blockquote>Yes, you need to know the joint distribution of $(X,Y)$ in order to compute the distribution of $Z$. That is, you need to know the probability that $Z \in [z