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Is a joint probability distribution between two X and Y random variables unique?

On the other hand, if we have a joint probability distribution, do we get the unique marginals X and Y?

Also, can we consider a joint probability distribution a kind of variance-covariance matrix between X and Y with the positive semidefinite property?

Your response is greatly appreciated!

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No, and it is quite easy to see why. If you think of plotting the marginals on two separate axis in 2-D space to obtain the joint, you can easily define any esoteric function which maintains their marginal distributions when brought back to the axis, yet has an arbitrarily defined joint distribution. And even with a defined covariance matrix it is not enough to get to uniqueness.

You need to define a copula function which will be able to uniquely define the marginals given the joint and vice versa. This is known as sklar's theorem.

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