This may come across as a stupid question, but I am confused as to the difference between a multivariate normal distribution and sampling multiple times from a single univariate distribution.
Lets say I get 2 i.i.d samples from a univariate gaussian distribution. Let the PDF of the univariate gaussian be $f_{x}$. Let these samples be the random variables $X_{1}$ and $X_{2}$. Consider the joint distribution of $X_{1}$ and $X_{2}$, given by the pdf $f_{X_{1},X_{2}} = f_{x}*f_{x}$. Is this a valid multivariate gaussian (with 0 correlation)?
I would appreciate any clarity you can give considering the two concepts.