I'm having some trouble wrapping my head around multi-variate distributions. Most textbooks describe it by starting off with two random variables [tex]X, Y[/tex] and introducing [tex]P(X \leq x, Y \leq y)[/tex]. This initially led me to believe that [tex]X, Y[/tex] uniquely determine the distribution over [tex]R^2[/tex] - I later confirmed with my TA that this isn't true and that one must explicitly specify the cdf on [tex]R^2[/tex]. 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.(adsbygoogle = window.adsbygoogle || []).push({});

Since this is the case, wouldn't it be better to describe [tex](X, Y)[/tex] as a map from the sample space to [tex]R^2[/tex] 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. [tex]Z = g(X, Y)[/tex]) and this is a bit confusing too. Don't I need the distribution on [tex](X, Y)[/tex] to get anything useful out of [tex]Z[/tex]?

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# Bivariate distributions

Can you offer guidance or do you also need help?

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