I know that if I know the marginal distributions, that's not enough to specify the joint distribution. But obviously it can't be "any" joint distribution, it still needs to respect its marginal requirements. So what do they look like?

For example, say I have the following "marginal requirements":

  1. $X \sim N(0, 1), Y \sim N(5, 10)$
  2. $\text{Corr}(X, Y) = 0.5$

Now, I know that 1 distribution for $(X, Y)$ satisfying these requirements is a multivariate normal distribution. But what are the other distributions? What is the "set" of distributions that can be used as a joint distribution for these marginal requirements? What do they look like? Do they have a specific "form"? What can I say about the density?

Are any of these questions answerable?

  • $\begingroup$ Item "2." is not a marginal requirement but a constraint on the joint. $\endgroup$ – Glen_b Dec 24 '18 at 0:13
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    $\begingroup$ You may like to investigate copulas, which covers the general case of joint distributions with known margins. Any copula which can reproduce the desired correlation when transformed back to normal margins will suffice, though. In some cases you could get pretty close to that correlation of 0.5 by choosing a Spearman correlation of around 0.5 on the copula. If you use R, two 'edge' examples can be seen like so: x=rnorm(1e6);y=ifelse(abs(x)>2.03,-x,x);z=ifelse(abs(x)>1.103,x,-x), where $ρ_{xy}≈ρ_{xz}≈0.5$ -- try plotting them. $\endgroup$ – Glen_b Dec 24 '18 at 0:25
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    $\begingroup$ Short answer; there are infinitely many different distributions: See this answer by Moderator cardinal for a pictorial catalog of some pretty ones. @Glen_b $\endgroup$ – Dilip Sarwate Dec 24 '18 at 3:07
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    $\begingroup$ As another example (related to one of the examples in Dilip's link), this is a large sample from a bivariate distribution (a Clayton(1) copula with normal margins) has very close to $\rho=\frac12$; indeed it may even be exactly 1/2 -- I haven't attempted to compute it -- but in any case some parameter very near 1 would have correlation 1/2. $\endgroup$ – Glen_b Dec 24 '18 at 4:53