Multivariate normal distribution vs. sampling multiple times from univariate normal distribution

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.

• Think about a multivariate normal that each component is normal but correlated to other components. Apr 14 at 15:39
• I would add that the pdf you suggest, using the product of the marginals, is a special case of the multivariate normal distribution. In this case, the covariance matrix is diagonal. Apr 15 at 0:53

Marginals are $$X,Y\sim N(0,1)$$. Draw from $$X$$ and $$Y$$ independently and get $$X = 1.5$$ and $$Y=-1$$. So far so good, right? Maybe the values are a little unusual and away from the means, but there's nothing too strange going on.
If $$X$$ and $$Y$$ have a strong positive correlation, then this pair of $$(1.5, -1)$$ is extremely unlikely. When $$X$$ is positive, $$Y$$ tends to be positive. Sampling only from the marginals means that this pairing is reasonably likely when it should not be.