Gaussian marginal vs Gaussian joint pdfs in simple terms

[blueraincap]

Can someone please attempt to explain the basic rationale of below statements as simply as possible?

- If (X,Y) follows a bivariate Gaussian distribution, X and Y follow univariate Gaussian distributions.
- X and Y follow univariate Gaussian distributions, (X,Y) does not follow a bivariate Gaussian distribution.

 

[guest_trader_1]

The probability of an event to happen that contains two variables implies that both variables have happened.

Independently if only one of them happened it does not imply that the other one will happen as well.

 

[globalarbtrader]

Can someone please attempt to explain the basic rationale of below statements as simply as possible?

- If (X,Y) follows a bivariate Gaussian distribution, X and Y follow univariate Gaussian distributions.
- X and Y follow univariate Gaussian distributions, (X,Y) does not follow a bivariate Gaussian distribution.

I think you mean (pendantic, but important):

- If (X,Y) follows a bivariate Gaussian distribution, X and Y follow univariate Gaussian distributions.
- X and Y follow univariate Gaussian distributions, (X,Y) does not have to follow a bivariate Gaussian distribution.

So if X and Y have a linear relationship that can be summarised with a corrrelation coefficient, and if they are both univariate Gaussian, then they are bivariate Gaussian.

But X and Y could be univariate Gaussian, but with some weird non linear relationship that isn't summarised with a correlation coefficient. Then they are not bivariate.

So we have eithier:

- At least one of X and Y are not univariate Gaussian, and regardless of their relationship they are not bivariate
- both X and Y are univariate Gaussian, but have some weird relationship that is non linear, and so they are not bivariate
- both X and Y are univariate Gaussian and have a linear relationship, so they are bivariate

It follows from the final statement that if X and Y are bivariate, they must also both be univariate. But we can have a situation where if X and Y are univariate, they are not neccessarily bivariate.


GAT

 

[blueraincap]

I think you mean (pendantic, but important):

- If (X,Y) follows a bivariate Gaussian distribution, X and Y follow univariate Gaussian distributions.
- X and Y follow univariate Gaussian distributions, (X,Y) does not have to follow a bivariate Gaussian distribution.

So if X and Y have a linear relationship that can be summarised with a corrrelation coefficient, and if they are both univariate Gaussian, then they are bivariate Gaussian.

But X and Y could be univariate Gaussian, but with some weird non linear relationship that isn't summarised with a correlation coefficient. Then they are not bivariate.

So we have eithier:

- At least one of X and Y are not univariate Gaussian, and regardless of their relationship they are not bivariate
- both X and Y are univariate Gaussian, but have some weird relationship that is non linear, and so they are not bivariate
- both X and Y are univariate Gaussian and have a linear relationship, so they are bivariate

It follows from the final statement that if X and Y are bivariate, they must also both me univariate. But we can have a situation where if X and Y are univariate, they are not neccessarily bivariate.


GAT

Yes, that's right. It suggests that univariate normality is a necessary but insufficient condition for bivariate normality. That is intuitive. So their bivariate pdf is given by 1. univariate pdf and 2. their copula.

 

[uscg2022]

Can someone please attempt to explain the basic rationale of below statements as simply as possible?

- If (X,Y) follows a bivariate Gaussian distribution, X and Y follow univariate Gaussian distributions.
- X and Y follow univariate Gaussian distributions, (X,Y) does not follow a bivariate Gaussian distribution.

Newly joined and was intrigued by your question so I thought I would try and break it out by
Adding (Bi) as bivariate Gaussian distribution AND (Ui) as univariate Gaussian distribution to represent it as a formula.

Assumptions:
(X,Y) equate to X and Y being together simultaneously, while (X) and (Y) equate to being seperate of each other.
If this assumption is True then the above statement can be represented as follows.

(X,Y) = [Bi] || (X) = [Ui] (Y) = [Ui]

(X) = [Ui] (Y) = [Ui] || (X,Y) ≠ [Bi]

(answer)
therefore if (X,Y) ≠ (Ui] and
(X) = (Bi) (Y) = [Ui] or (X) = [Ui] (Y) = [Bi]

* The variables are mutually exclusive when they are following different Gaussian distributions.

** I took a shot, hope it helps. ;-)