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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.

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    $\begingroup$ Think about a multivariate normal that each component is normal but correlated to other components. $\endgroup$
    – B.Liu
    Commented Apr 14, 2021 at 15:39
  • $\begingroup$ 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. $\endgroup$
    – PedroSebe
    Commented Apr 15, 2021 at 0:53

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What you propose ignores correlations. When marginals have correlation (say positive for now), then when one is high, the others will tend to be high.

Example

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.

If the marginals are independent of one another, then what you propose is fine.

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