# Joint distribution given marginals and correlation

I am trying to understand whether it is possible and if so then how to find a joint distribution given marginals and a correlation matrix.

In particular, suppose $$X_{1},\ldots,X_{n}$$ are discrete random variables. I am given their marginals $$p_{X_{1}},\ldots,p_{X_{n}}$$ and their correlation matrix. Can I construct (uniquely?) the joint $$p_{X_{1},\ldots,X_{n}}$$?

I tried this for $$n=2$$, assuming each of the two random variables has two possible outcomes $$a$$ and $$b>a$$, and assuming that the marginals of $$X_{1}$$ and $$X_{2}$$ are the same given by $$\mathbb{P}[X_{1}=a]=c$$. Let $$r$$ be the correlation of $$X_{1}$$ and $$X_{2}$$. My Mathematica readily calculates unique joint distribution provided $$c(1-r)<1$$ and claims there is no joint otherwise.

How would I generalize to more random variables with more than two possible outcomes? What I do in Mathematica (calculate correlation from joint distribution and then solve for the joint distribution that gives $$r$$) does not generalize nicely. Also, are there some results that would tell me what I am doing can/cannot be done? Uniquely?

I tried reading StackExchange questions/answers here, here, here, here, and here. But none of the answers help me forward. Also, I know that this question is trivial when the random variables are independent. So let's disregard that special case.

In general, the joint distribution is not uniquely determined by the marginals. The correlations only give second order moment information so are still not enough to pin down a unique joint distribution.

Adapting this example from wiki, suppose $$W$$ is a random variable that takes values $$-1,1$$ with equal probability, and $$X\sim N(0,1)$$. Assume $$X$$ is independent of $$W$$. Define $$Y=WX$$. You can show $$X,Y$$ are uncorrelated and are marginally standard normal but not independent. However, two iid standard normals are also uncorrelated.

Here is a discrete example with infinitely many joint distributions that yield the same uncorrelated marginal distributions:

Suppose $$X\in \{-1,0,1\}$$ and $$Y\in \{0,1\}$$. For some $$p\in (1/6,1/3)$$, consider the joint distribution \begin{align} P(X=-1,Y=1)&=P(X=1,Y=1)=p,\\ P(X=-1,Y=0)&=P(X=1,Y=0)=1/3-p,\\ P(X=0,Y=0)&=2p-1/3\\ P(X=0,Y=1)&=2/3-2p\\ \end{align}

The marginals are given by

$$P(X=x)=1/3 \quad \forall x\in \{-1,0,1\}\\ P(Y=0)=1/3,P(Y=1)=2/3$$

and $$X,Y$$ are uncorrelated (note $$X$$ has mean zero):

$$E[XY]=(-1)(1)p+(1)(1)p=0.$$

• Thank you, this is insightful. Yes, correlation pins down only the second order. But this is very heuristic. The example is beautiful. However, I do not understand how $Y$ is standard normal. In the example, $a=1$, so yes. But for general $a$, the variance of $Y$ is $a^{2}$ if I am correct? Moreover, I have not been able to find any formal (book) treatment of the issues I am asking about. And if not results, then (counter-)examples. Maybe because questions about uniqueness/existence of a joint given marginals and other conditions seems to have distribution/parameter-specific answers.
– Jan
Commented Mar 23, 2023 at 20:16
• @Jan You are right! Thanks for pointing out! I have updated the post to show a less ambitious counterexample, namely that the same marginals and correlation of zero can produce at least two joint distributions; also, the wiki link has another "asymmetric example" of uncorrelated standard normals. You may also be interested in copulas. Commented Mar 23, 2023 at 20:55
• Thank you. I noticed copulas in some of my reading but did not pay much attention assuming that they apply to continuous r.v.s and I am mainly interested in discrete ones. I will keep educating myself.
– Jan
Commented Mar 23, 2023 at 21:12
• @Jan I added a discrete example with infinitely many joint distributions--hope that helps. Commented Mar 23, 2023 at 22:57
• The discrete example is great, and proves that my ''quest for uniqueness'' is bound to fail. Thanks for proving me wrong.
– Jan
Commented Mar 24, 2023 at 13:56