I am trying to find the joint probability distribution of four variables. These variables are all separately normally distributed with mean 0 and variance 1.
I've been looking at various correlation statistics between these variables which are as follows:
Expectation of all products of pairs and triples of different variables = 0
Expectation of product of all four variables = 1
My first instinct was that, since the covariance matrix is diagonal, this would just be a set of simple gaussians. Perhaps this is true however it seems not to be from other posts about bivariate normals and my intuition regarding the context of the problem.
Is there a general route/methodology I should be looking into in order to recreate the joint distribution? Or is this not a well posed question?
Happy to provide more context if it would help, apologies, I'm not a frequent poster!
If $X_1,X_2,X_3,X_4$ are jointly normal with each of them having mean $0$ and variance $1$ and if the covariance matrix is diagonal then the joint density is $f(x_1,x_2,x_3,x_4)=\frac 1 {(2\pi)^{2}} e^{-(x_1^{2}+x_2^{2}+x_3^{2}+x_4^{2})/2}$. However if you just know only that each of them is normally distributed and the covariance matrix is diagonal you cannot find the joint distribution.